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THE THEORY OF 

ELECTROMAGNETIC 

WAVES 



A S YMPOSIUM 



Held under the Auspices of 

THE WASHINGTON SQUARE COLLEGE OF ARTS AND SCIENCE 
and the INSTITUTE FOR MATHEMATICS AND MECHANICS 

OF NEW YORK UNIVERSITY 

and the GEOPHYSICAL RESEARCH DIRECTORATE OF THE 

AIR FORCE CAMBRIDGE RESEARCH LABORATORIES 

June 6-8, 1950 




19 5 1 
INTERSCIENCE PUBLISHERS, INC. 



Published simultaneously in Communications on Pure and Applied Mathematics, A Journal 
Issued Quarterly by the Institute for Mathematics and Mechanics of New York University, 
Volume III, No. 4, pages 355-449 (1950) ; Volume IV, No. 1, pages 33-160, and No. 2/3, pages 
225-378 (1951) 



LIBRARY OF CONGRESS CATALOG CARD NUMBER 51-14016 

ALL RIGHTS RESERVED. This book or any part thereof 
must not be reproduced in any form without permission 
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FOREWORD 
BY RICHARD COURANT 

Director, Institute for Mathematics and Mechanics, New York University 

With the rise of modern atomic physics, the interest of physicists in the 
classical theory of electromagnetism, that is, the theory based on Maxwell's 
equations, waned and the field was all but left to the mathematicians. At first, 
brilliant contributions were made by men such as H. Poincare, Lord Rayleigh, 
Arnold Sommerfeld, Hermann Weyl, and G. N. Watson, who gave mathematical 
answers to a number of important problems of electromagnetic wave propaga- 
tion. But then mathematicians too began to lose interest in a field that was 
becoming less and less fruitful. 

At this time, when neither physicists nor mathematicians were interested in 
classical electromagnetism, a great number of new problems were suggested by 
engineers. Progress in the technique of the radio made it possible to utilize higher 
frequencies; further knowledge was needed of wave guide and cavity theory, 
diffraction and reflection of high-frequency waves, propagation in non-homog- 
eneous atmospheres, antennas, and new principles of microwave generation. 

The demands of industry coupled with military needs finally compelled the 
attention of scientists. The Radiation Laboratory in Cambridge was set up to 
attack electromagnetic problems systematically. By a new approach, rather 
than by the ingenious use of old tools, members of this laboratory — notably 
J. Schwinger and his associates — obtained far-reaching practical and theoretical 
results; for these results they used Green's functions, integral equations, the 
Wiener-Hopf procedure, variational techniques, and asymptotic methods. Math- 
ematicians, stimulated by their success, began once again to consider electro- 
magnetic theory a worth while field of endeavor. Even old problems, such as 
the influence of the ionosphere on wave propagation, a problem still incompletely 
solved, were attacked. 

Those who attended the Symposium on the Theory of Electromagnetic 
Waves held at New York University in June of 1950 know how much the Geo- 
physical Research Directorate of the Air Force Cambridge Research Laboratories 
has done to support and encourage this revived interest. 

In publishing the Proceedings of the Symposium, New York University and 
the Geophysical Research Directorate expect to increase the usefulness of the 
meetings. Since the beginnings of Maxwell's work in electromagnetic theory the 
subject has been eminently mathematical; there is thus some measure of justifi- 
cation in the trend to leave much of its development to mathematicians. It is 
hoped that the examples of research appearing in the present publication will 
encourage other mathematicians to cooperate wdth engineers and physicists in 
this fascinating field. 



FOREWORD 

BY Lt. Colonel F. C. E. ODER 

Geophysical Research Directorate, Air Force Cambridge Research Laboratories* 

It is with great pleasure that the Geophysical Research Directorate of the 
Air Force Cambridge Research Laboratories co-sponsored the Syraposium on the 
Theory of Electromagnetic Waves and greets the publication of its Proceedings. 
As a part of its deep conviction for the necessity of long-term research, the 
Directorate welcomed the opportunity to encourage and accelerate research in 
electromagnetic theory, an understanding of which is so essential for all types of 
radio communication. 

Indeed, the Department of the Air Force is confronted with many problems 
in wave propagation. In view of the tremendous range of present-day aircraft 
and the still greater range and altitude of future airborne vehicles, electromagnetic 
propagation around a spherical semiconductor surrounded by a non-concentric 
anisotropic inhomogeneous reflector must be considered at both large and small 
wavelengths. 

However, we fully realize the necessity for studying fundamentals before 
the solutions of specific problems arising in radio communication may be obtained. 
Both the Geophysical Research Directorate and the Institute for Mathematics 
and Mechanics recognize the importance of research as distinct from development. 
This union of the sponsoring agencies represents on the part of each an expression 
of faith in the system whereby freedom of science and research must and mil 
prevail. 

A prime purpose in holding this Conference is the exchange of information 
within the field of electromagnetic waves; the cross-fertilization of scientific 
ideas in such a conference strengthens both mathematical physics and electro- 
magnetics. In fact, the principle of the cross-fertilization of ideas is productively 
applied internally within the Geophysical Research Directorate. Thus, the 
Directorate comprises five laboratories, each dealing with a slightly different 
aspect of geophysics. These are the Atmospheric Analysis Laboratory, the 
Upper Air Laboratory, the Terrestrial Sciences Laboratory, the Atmospheric 
Physics Laboratory, and the Atmospheric Ionization Laboratory (formerly called 
the Electromagnetic Propagation Laboratory). Practices and theories used in 
one branch of geophysics are brought to the attention of investigators engaged 
in other branches of the subject. The broad field covered by the Geophysical 
Research Directorate includes theoretical and experimental investigations on the 
dynamics and structure of the atmosphere, the hydrosphere, and the lithosphere, 
and the relationships and energy transference among them. The various prob- 
lems in each of the five laboratories mainly concern the atmosphere inasmuch as 



•^Now known as the Geophysics Research Division, Air Force Cambridge Research Center. 



FOREWORD V 

it is in this region that the Air Force operates. However, the effects occurring 
within the earth and the seas must also be considered because of their effect 
upon the lower atmosphere. 

With such a broad attack on the planet as a whole, the techniques used 
in any one laboratory necessarily employ many scientific disciplines, including 
as a minimum mathematical physics, chemistry, physical chemistry, and mathe- 
matics. Only by utilizing all these various disciplines is it possible to undertake 
an adequate study of the mechanics and phenomena of the various divisions of 
the terrestrial atmosphere: the troposphere, stratosphere, chemosphere, ion- 
osphere, mesosphere, and exosphere, the latter region extending to the boundary 
with interstellar space. 

It is largely through the efforts of the Atmospheric Ionization Laboratory 
that this Symposium was initiated. Because this Laboratory employs electro- 
magnetic waves as a probing tool, it must know the behavior of these waves in 
media of various types. This Laboratory uses, for example, very long radio 
waves to investigate the properties and characteristics of the lower ionosphere, 
and microwaves to examine such tropospheric mechanisms as the physics of 
clouds and precipitation. Similarly, the infrared and higher frequencies of the 
electromagnetic spectrum are utilized for a study of the ionosphere and mesosphere. 

It is obvious from even this very brief indication of the activities of the Geo- 
physical Research Directorate of the Air Force Cambridge Research Laboratories 
that many problems in mathematical physics have yet to be considered. Re- 
search on these problems will undoubtedly lead to the formation of new tools for 
the investigation of the physics of the atmosphere. 

In conclusion, I should like to express our thanks to the distinguished group 
of scientists who contributed to this Symposium and especially to those who 
crossed oceans and national borders to participate. 



ACKNOWLEDGMENT 

The Symposium on Electromagnetic Waves was sponsored jointly by the 
Geophysical Research Directorate, Air Force Cambridge Research Laboratories, 
and the Institute for Mathematics and Mechanics, New York University. Since 
recent advances in electromagnetics have been both rapid and important, and 
since no conference on this subject has been held for some years, the meeting 
was set for June, 1950. The conference was intended to bring theoreticians 
abreast of late developments in electromagnetics; problems still outstanding 
could be reviewed; and applied problems of particular urgency could be discussed. 
The sponsors believed that the stimulus provided by personal contacts, open 
discussion, and critical examination of current problems would markedly ad- 
vance research already in progress. 

The Symposium was conceived in the Atmospheric Ionization Laborator}^ 
of the Geophysical Research Directorate. Arrangements were made by a com- 
mittee consisting of M. Kline of New York University and N. C. Gerson of the 
Geophysical Research Directorate, assisted by Professors S. G. Roth, B. Fried- 
man, F. W. John, H. E. Wahlert, and Messrs. L. Kraus, J. Schmoys, and J. 
Lurye of New York University. Without their careful planning the S^nnposium 
could scarcely have come into being. That the Symposium was a success may 
be attributed solely to the cooperation, participation, and enthusiasm of the 
conferees. 

Fredekic C. E. Oder 
Lt. Colonel, USAF 
Director, Base Directorate for 
Geophysical Research 
Air Force Cambridge Research 
Laboratories 



THE THEORY OF ELECTROMAGNETIC WAVES 



CONTENTS 

Foreword, by R. C. COURANT iii 

Foreword, by Lt.-Col. F. C. E. ODER v 

Acknowledgment vii 

On the Theory of Electromagnetic Wave Diffraction by an Aperture in an 
Infinite Plane Conducting Screen, 

hy H. LEVINE and J. SCHWINGER 1 

On Systems of Linear Equations in the Theory of Guided Waves, 

hy W. MAGNUS and F. OBERHETTINGER 39 

Wiener-Hopf Techniques and Mixed Boundary Value Problems, 

hy S. N. KARP 57 

Asymptotic Solutions of a Differential Equation in the Theory of Microwave 
Propagation, 

hy R. E. LANGER - 73 

Criteria for Discrete Spectra, 

hy K. O. FRIEDRICHS 85 

Extension of Weyl's Integral for Harmonic Spherical Waves to Arbitrary 
Wave Shapes, 

hy H. PORITSKY 97 

Kirchhoff's Formula, Its Vector Analogue, and Other Field Equivalence 
Theorems, 

hy S. A. SCHELKUNOFF 107 

On the Diffraction Theory of Gaussian Optics, 

hy H. BREMMER 125 

Diffraction and Reflection of Pulses by Wedges and Corners, 

hy J. B. KELLER and A. BLANK 139 

Vector Waves Functions, 

hy R. D. SPENCE and C. P. WELLS 159 

The W.K.B. Approximation as the First Term of a Geometric-Optical Series, 

hy H. BREMMER • • • 169 

Remarks Concerning Wave Propagation in Stratified Media, 

hy S. A. SCHELKUNOFF 181 

vii 



VIU CONTENTS 

The Theory of Magneto Ionic Triple SpUtting, 

hy 0. E. H. RYDBECK 193 

An Asymptotic Solution of Maxwell's Equations, 

hy M. KLINE 225 

Field Representations in Spherically Stratified Regions, 

hy N. MARCUVITZ 263 

Propagation in a Non-homogeneous Atmosphere, 

hy B. FRIEDMAN 317 

Reflection of Electromagnetic Waves from Slightly Rough Surfaces, 

hy S. 0. RICE 351 

The Theory of Scattering of Radio Waves in the Troposphere and Ion- 
osphere (abstract) y 

hy H. G. BOOKER 379 

Properties of Guided Waves on Inhomogeneous Cylindrical Structures 
(ahstract) , 

hy R. B. ABLER 381 

Evaluation of Integrals Associated with Wave Motion in Dispersive ]\Iedia 
and the Formation of Transients (ahstract), 

hy M. CERRILLO 383 

Electromagnetic Research in the U. S. Air Force Research Program, 

hy N. C. GERSON 389 



On the Theory of Electromagnetic Wave Diffrac- 
tion by an Aperture in an Infinite Plane 
Conducting Screen 



By HAROLD LEVINE and JULIAN SCHWINGER 
Lyman Laboratory of Physics, Harvard University 

1. Introduction 

The diffraction of electromagnetic and light waves by an aperture in a 
plane conducting screen is a classical boundary value problem. As is well known, 
theoretical analysis aims at a solution of the vector Maxwell equations, which 
incorporates a prescribed form of excitation and satisfies appropriate boundary 
conditions on the screen and in the aperture. 

A small measure of progress towards this objective results from the Kirchhoff 
diffraction theory, which identifies aperture and incident fields and arbitrarily 
assigns null values to the field components on the shadow face of the screen. 
The Kirchhoff formulation suitable for an electromagnetic field (assuming har- 
monic time variation) is given by Stratton and Chu [1]; this includes charge 
distributions on the rim of the screen to ensure that the free space fields obey 
the Maxwell equations. A defect in the Kirchhoff procedure is revealed by its 
failure to duplicate the assumed boundary values at the conducting screen. 
The lack of self-consistency has a further consequence that Kirchhoff predictions 
are qualitatively correct only if the wave length of the electromagnetic field is 
small in comparison with all aperture dimensions, for then the field on the 
shadow face of the screen is relatively small. 

Another method of analysis, which provides information at long wave 
lengths, is due to Lord Rayleigh [2]. The basic idea is that, in the vicinity of 
the aperture, the electromagnetic field distributions can be calculated as though 
the wave length were infinite, making available the results of potential theory. 
As an example, Rayleigh treats the case of a circular aperture, with normally 
incident harmonic plane waves [3]. After identifying the local field with that of 
a Hertzian oscillator, the known radiation characteristics of the latter are used 
to find the diffracted field at large distances from the aperture. The tangential 

Paper presented at the June, 1950, Symposium on the Theory of Electromagnetic Waves, under 
the sponsorship of the Washington Square College of Arts and Sciences and the Institute for 
Mathematics and Mechanics of New York University and the Geophysical Research Directories 
of_^the^Air Force Cambridge Research Laboratories. 

355 (SI) 



356 (S2) ELECTROMAGNETIC WAVES 

electric field at the screen vanishes in this solution, as required by the boundary 
condition on a perfectly conducting surface. However, the predicted transmis- 
sion cross section (which measures the ratio of energy passing through the 
aperture per second to that transported per unit area of the incident wave) is 
accurate for long wave lengths only, representing the first term of an expansion 
for this quantity in ascending powers of the ratio, (radius of aperture/wave 
length). Bethe [4] considers Rayleigh's example again, and extends the theory 
to apply for an arbitrary spatial incident field. A new feature is the diffracted 
field representation by fictitious magnetic charges and currents in the aperture; 
their low frequency distributions are obtained, having due regard for all boundary 
conditions. The resulting plane wave transmission cross sections are accurate 
to the same order of approximation as in Rayleigh's theorj^ 

A procedure for obtaining an exact solution to this problem, valid at all 
wave lengths, is described recently by Meixner [5]. The anabasis is carried out 
with a pair of scalar electromagnetic potentials, akin to those which Debye [6] 
employed in the theory of diffraction by a spherical obstacle. In addition to 
requirements imposed by the wave equation, boundary and radiation conditions, 
Meixner prescribes supplementary conditions for the potentials. It is the pur- 
pose of the latter conditions, enforced at the rim of the screen, to assure quad- 
ratic integrability there for the electromagnetic field components. Indeed, the 
guarantee of a finite electromagnetic field energy in any arbitrarily small region 
of space is regarded as an essential feature of a unique and physicalh" acceptable 
solution [7]. For explicit construction of the potentials, spheroidal coordinates 
are appropriate, as these permit a convenient description of the circular aperture, 
and allow separation of variables in the wave equation. INIeixner obtains infinite 
series expansions for the potentials in terms of spheroidal functions, although 
numerical evaluation is deferred. In this connection, it may be anticipated 
that slow convergence of the series with increasing frequency- will render com- 
putation difficult. 

From the brief survey of theoretical methods available for three dimensional 
electromagnetic diffraction problems, the need for approximation procedures, 
accurate in a large frequency range, is evident. The latter would be particularly 
appropriate for problems which do not admit of analysis in terms of known 
solutions to the wave equation. This paper, a sequel to previous ones concerned 
with diffraction in a scalar field [8], describes the nature of variational principles 
for obtaining some of the desired information [9]. ' 

The general features of the investigation which follows pertain to the steady 
Btate diffraction problem for an aperture of arbitrary shape in a perfecth' con- 
ducting screen, with incident plane electromagnetic waves. 

A formal description of the fields on opposite sides of the screen, and a 
schedule of boundary conditions in the plane of the latter is given, utilizing 
symmetry properties of the Maxwell equations with respect to reflection in a 
plane. To apply these boundary conditions, expressions for the field vectors 



HAROLD LEVINE AND JULIAN SCHWINGER (S3) 357 

within any region are derived in terms of the tangential components of either 
electric or magnetic fields on the boundary of the region. The mathematical 
tools for exhibiting such relations are tensor (or dyadic) Green's functions, whose 
properties are briefly described. On the far (shadow) side of the screen, the 
electric and magnetic field at any point can be represented by surface integrals 
involving the tangential electric aperture field; similar expressions apply on the 
near side of the screen, along with the incident and reflected fields appropriate 
to a completely infinite screen. The electromagnetic field thus constructed 
satisfies Maxwell's equations at all points of space, and moreover its tangential 
electric component vanishes at the screen and is continuous through the aperture. 
From equality of the respective tangential magnetic fields in the aperture, or 
equivalently, of the transmitted and incident components, an integral equation 
for the tangential electric aperture field is obtained. Employing the integral 
equation (whose solution is seldom feasible), a stationary property of the spherical 
wave radiation field at large distances from the aperture is established, subject 
to small independent variations (about the correct values) of the tangential 
electric aperture fields due to a pair of incident waves. 

Alternatively, electric and magnetic fields on the far side of the screen are 
uniquely determined by values of the tangential magnetic field at the shadow 
face of the screen and in the aperture (the latter being equal to those of the 
incident magnetic field). An integral equation to determine the magnetic field 
distribution on the screen is a consequence of the null value there for the tan- 
gential electric field. The integral equation can be utilized to obtain another 
stationary property of the radiation field, which involves the distributions 
arising from a pair of incident waves. 

A closely related variational principle is based on description of the field in 
terms of the current on (or the discontinuity in tangential magnetic field at) 
the screen. The electric and magnetic field at any point of space can be indi- 
vidually represented by a surface integral containing the current, to which the 
corresponding incident field is added. These fields satisfy the Maxwell equations 
and exhibit continuous variation in passing through the aperture; the require- 
ment of vanishing tangential electric field at the screen yields an integral equa- 
tion to specify the current distribution, from which the variational principle is 
constructed. 

The plane wave transmission cross section of the aperture shares these 
stationary properties, based on a theorem which relates the cross section to the 
imaginary part of the radiation field amplitude in the direction of incidence. 
Complementary aspects are exhibited by the different forms of cross section, 
with low frequency behavior readily accessible to aperture electric field approxi- 
mations, and high frequency behavior to current approximations. In general, 
the overall agreement of numerical results obtained from the variational formu- 
lations allows an estimate of proximity to the correct solution. 

The variational method is applied in detail to the problem of diffraction by 



358 (S4) 



ELECTROMAGNETIC WAVES 



a circular aperture, with normally incident plane waves. Numerical results for 
the transmission cross section are compared with those yielded by the Kirchhoff 
and Rayleigh approximations. 

2. Formulation of Boundary Value Problem 

We consider an infinitesimally thin, perfectly conducting plane screen S2 , 
of infinite extent, which is perforated by an aperture Si , and located in other- 
wise empty space. A rectangular coordinate system is chosen with origin at 
some point of the aperture, and oriented so that the screen Ues in the a:,2/-plane 
(Figure 1). 




Fig. 1. Diffracting aperture in a plane screen. 



A plane electromagnetic wave is incident on the aperture in the half space 
z < 0; it is desired to investigate the diffracted field. The incident wave, with 
propagation vector n' and polarization vectors e', h', is described by 



(2.1) 



HAROLD LEVINE AND JULIAN SCHWINGER (S5) 359 

E'^Xr) =e'exp {ikn'-r} 

— e' exp {ik{x sin ??' cos <p^ + y s'm §' sin (p' -^ z cos ^') ] 
H'-^Xr) =h'exp [iW-x}, 



e' = h' X n', h' = n' X e', e'-e' = h'h' = n'n' = 1, 

where k = 2t/X is the wave number and X the wave length. The harmonic 
time dependence, exp {—ikd], with c the velocity of wave propagation ( = 
3.10^^cm/sec), is omitted throughout. 

For the complete (incident + diffracted) field, the electric and magnetic 
intensities are governed by the free space Maxwell equations (Gaussian units 
are employed) 

V X E = ikU, V-E = 
(2.2) 

V X H = -ikE, V-H = 

and subject to the boundary condition 

(2.3) e, X E = 0, r on ,S2 

where e^ is a unit vector in the z direction; both electric and magnetic fields 
vary continuously through space, including traversal of the aperture. 

The diffraction problem may be formulated in different ways, according to 
the nature of the field existence theorem employed. In addition, there are 
alternate geometrical viewpoints, with counterparts in mathematical formula- 
tion. For one, the aperture is obtained by excising part of a completely infinite 
screen, and becomes a couphng surface for the half spaces on opposite sides of 
the screen; the other regards the screen as an obstacle inserted in free space. 
Although equivalent results are obtained by the diverse procedures if the problem 
is treated rigorously, these lead to independent, complementary variational 
principles useful in the approximation sense. 

Considerations of symmetry provide information about the fields on opposite 
sides of the screen and their relation in the aperture. Let us write 

E(r) = Eo(r) + E,(r), H(r) = Ho(r) + H,(r), z < 
(2.4) 

E(r) = E,(r), H(r) = H,(r), z > 

where Eo(r), Ho(r) describe the field in the absence of an aperture, 

(h (r)) ^ (hV ^^P {^"^'t} "F (]^.)'(^ - 2e,eJ exp {f/bn'-(r - 2e.TeJ} 

e. XEo = 0, e,-Ho = 0, z = 0, 



360 (S6) ELECTROMAGNETIC WAVES 

Before subjecting (4) to the boundary conditions at the plane z = 0, it is con- 
venient to classify solutions of the Maxwell equations according to their sjin- 
metrj^ in the z coordinate. The even and odd solutions with respect to reflection 
in the plane 2; = are 

Et{x, y, z) = :hE,(x, y, -z), Ht{x, y, z) = "THtix, y, -z) 
(2.6) 

EXx, y, z) = "TEXx, y, -z), H^x, y, z) = ±HXx, y, -z) 

respectively, where Et , Ht signify components transverse to the z direction, 
i.e. tangential to the x,y-plsine. 

Each odd solution, whose tangential electric field components vanish in 
the aperture as well as upon the screen, describes a field configuration with the 
plane z = completely occupied by a perfect conductor. The fields Eo , Ho 
constitute an odd solution, resulting from superposition of the incident plane 
wave and a plane wave specularly reflected from the conducting surface. Owing 
to the geometrical identity of the half spaces z ^ 0, the fields attributed to the 
presence of an aperture, Ei,2 , Hi, 2 , belong to the class of even solutions, ^dz: 

E^tix, y, z) = Ertix, y, -z), H2t(x, y, z) = -H^ix, y, -z) 

(2.7) 

E2^{x, y, z) = -E,Xx, y, -z), H2^{x, y, z) = H.X^, V, -2). 

Accordingly, the boundary conditions relating to the tangential components of 
(4), 

E2t = Eit , H2t — Hit = Hot 
(2.8) 

E2t = = Eit , 

are satisfied if E2t vanishes on the screen, and 

(2.9) H2t = Wot = Hr, 

The boundary conditions for normal components need not be considered ex- 
plicitly, as these are automatically satisfied if the conditions for the tangential 
components are fulfilled. We note, in particular, that 

(2.10) E2. = iEo. , r in ;Si . 

If the screen were regarded as an obstacle to the propagation of the incident 
wave through free space, we could write 

E(r) = E^^Xr) + E(r) 
(2.11) 

H(r) = W'^Xr) + H(r) 

everywhere, subject to the boundary condition (3). 



^Ot } 


r 


in 


Si 




r 


on 


S2 


r in 


s. 







HAROLD LEVINE AND JULIAN SCHWINGER (S7) 361 

3. Tensor Greenes Functions 

Our next task is the formulation of explicit boundary value problems in 
accordance with the general field requirements outlined above. For this pur- 
pose, we make use of the well known existence theorem that the fields within 
a region are uniquely determined by the values of the tangential components 
of the electric field, or the magnetic field, on the bounding surface of the region. 
In order to exhibit this relation explicitly, the concept of the Green's function 
is introduced. The type of Green's function required is somewhat more general 
than that usually encountered, for we desire to obtain a linear relation between 
the field vectors within a region and the field vectors on the surface of that 
region. Accordingly, the coefficients in that relation must be of the character 
of tensors, or dyadics. The remainder of this section is devoted to the theory 
of the tensor Green's functions, and the derivation of field representations; our 
account follows closely the M.I.T. Radiation Laboratory Report 43-34(1943) 
by J. Schwinger. 

The electromagnetic fields within a region occupied by both electric and 
magnetic charge and current are described by 

V X E = ^m - — J*, V X H = -ikE + — J 
(3.1) ^ "^ 

V-E = 47rp, V-H = 47rp* 

where p(r) and J(r) are the electric charge and current densities, and p*(r), 
J*(r) represent the analogous magnetic quantities; the time dependence of all 
quantities is exp { —ikct]. The electric and magnetic fields, individually, obey 
the equations: 



V X (V X E) -k'E = ^^ J - ^ V X J^ 



(3.2) 



c c 



V X (V X H) - k'n = ^^ J* + ^ V X J. 

c c 



We shall define the tensor Green's functions associated mth a region V 
bounded by a surface S, in terms of the field which a point current would produce 
within the region if it were enclosed by perfectly conducting metallic walls, 
coinciding ^^dth the surface S. Consider, therefore, the electric field produced 
by an electric current density 

(3.3) J(r) =e5(r-rO 

where e is an arbitrary constant vector, and 5(r — r') is defined by 



362 (S8) ELECTROMAGNETIC WAVES 

(3.4) I 5(r - rO dr =1, 5(r - r') = 0, | r - r' | ?^ 

in which the integration is to be extended over a region enclosing the point r'. 
The components of the electric field will evidently be linearly related to the 
components of the constant vector e, and we therefore write, in dyadic notation, 

(3.5) E(r) = ^ r'"(r, r')-e = ^^ [ r<"(r, r") -JCr") dr" 

C C J 

upon which is imposed the boundary condition that the tangential components 
of E vanish at the surface S, or 

(3.6) n X E(r) = 0, r on >S 

where n is the outwardly drawn normal to the surface S at the position r. We 
have thereby introduced the dyadic r^^^(r, r'), which we shall call the electric 
field Green's function, defined by 

V X (V X r^^^(r, rO) - A;'r^^^(r, rO = £5(r - r') 
(3.7) 

n X r^'^(r, r') = 0, r oti S 

where e represents the unit dyadic. Similarly, the magnetic field produced by 
the magnetic current density 

(3.8) J*(r) = e6(r - r') 
can be written 

(3.9) H(r) = ^^ r^'^(r, rO-e = ^^ f r''\T, r'0-J*(O dr''. 

c c J 

However, here the boundary conditions do not relate directly to the mag- 
netic field, but rather to the accompanying electric field, and thus state: 

(3.10) n X (V X H(r)) = 0, r on S. 
The equations 

V X (V X r^^^(r, r')) - k'r''\T, rO = £5(r - r') 
(3.11) 

n X (V X r^^^(r, rO) = 0, r on 5 

therefore define a second dyadic; r^^^r, r'), which we shall term the magnetic 
field Green's function. 

For infinite empty space, devoid of metallic objects, there is no distinction 
between electric and magnetic field Green's functions. The fundamental tensor 
Green's function of free space, r^°\r, r'), which is a solution of the differential 



il 



HAROLD LEVINE AND JULIAN SCHWINGER (S9) 363 

equation (7) or (11), and describes a spherical wave moving outwards at large 
distances from the source point, appears in the closed form (see Appendix 1): 

(3.12) r'»'(r, r') = (e - p Vv) - "J"', J L^^', " = r'°'(r', r). 

The tensor Green's functions for a half space, with an infinite plane conduct- 
ing boundary, are easily constructed. For either of the current densities (3) or 
(8), the fields are those in the absence of a conducting boundary, provided a 
suitably disposed image current is introduced. From this scheme, we find 



(3.13) ri.^^'^'^(r, rO = r^°^(r, r') =F r^°^(r, r' - 2e,e,T0-(£ - 2e.eJ, z,z' > 

the upper and lower signs to be employed for r+\ r+^ respectively. By way 
of verification, observe that the expression for r+ ^ provides vanishing tangential 
electric field components at the conducting boundary (z = 0), whereas the 
normal component is double its free space value; all in accord with the boundary 
conditions. 

In the following development, use is made of a general vector relation 
between surface and volume integrals, 

f dSn-[B X (V X A) - A X (V X B)] 
(3.14) 

= f dr [A-V X (V X B) - B- V X (V X A)] 

which is termed Green's second vector identity. As a first indication of its 
usefulness, we can show that the r's share the fundamental symmetry property 
of all Green's functions: 

(3.15) r(r',r") = [r(r'', r')]'' 

(3.16) V X r^^^(r', r'O = [V" X r^'^(r'', r')]'' 

where r^ denotes the transposed dyadic r^ = r^ . Equation (15) is estab- 
lished by applying Green's second vector identity to the functions A(r) = 
r(r, r') -e', B(r) = r(r, r'O •©'', in which e' and e'' are arbitrary constant vectors 
and the r's can be either electric or magnetic field Green's functions. The 
relation (16) is readily obtained from (14) by substituting 

A(r) = V X r^'^(r, r^-e', B(r) = r^^^(r, r'O-e". 

The fields of physical interest are those contained within regions devoid 
of charge and current (equations 2.2) and both electric and magnetic fields are 
thus required to satisfy the vector wave equation: 



364 (SIO) ELECTROMAGNETIC WAVES 

V X (V X E) - A;'E = 
(3.17) 

V X (V X H) - /b'H = 0. 

Consider now a region V bounded by a surface S', which is contained 
Tsnthin V, the region of definition of the Green's functions, and which may, in 
particular, coincide with it. We wish to express the fields within V in terms 
of the tangential field components on the boundary surface *S'. To this end, 
let us employ Green's second vector identity, with 

A(rO = E(rO, B(rO = r^^^(r', r)-e. 
We obtain 

- [ dS'n'-[im(T') X (r"^(r', r).e) + E(rO X (V X r^^'(r', r)-e)] 



I 



(^T'E(rO-e5(r' - r) = E(r)-e 



if the point r is contained within the region V; otherwise the volimie integral 
vanishes. Therefore the electric field within the region V is related to the 
tangential components of the electric and magnetic fields on the surface S' by 

E(r) = -ik [ dS' (n' X H(rO)-r^^^(r', r) 

J S' 

(3.18) 

- f dS' (n' X E(r'))-(V' X r'"(r', r)). 

J S' 

The physical interpretation of this result can be made more apparent by 
employing the theorems (15), (16) to rewrite (18) in the form 

E(r) = -ik f r^'^(r, rO-(n' X H(rO) dS' 

J S' 

(3.19) 

- V X f r"'(r,r')-(n' X E(r')) dS'. 

J S' 

Recalling the definitions of the Green's functions in terms of the fields of point 
currents (equations (5) and (9)), we observe that (19) is just the electric field 
which would be produced by an electric surface current density 

(3.20) K(r) = -|^nXH(r), 
and a magnetic surface current density 

(3.21) K*(r) = -f n X E(r) 



HAROLD LEVINE AND JULIAN SCHWINGER (Sll) 365 

located on the surface S' (with outward normal n). If, therefore, at the same 
time the actual field is removed, and the surface currents (20), (21) are caused 
to flow, the field within V will remain unaltered, while the field outside V will 
be reduced to zero. 

When the surface S' coincides Avith S, the first integral in (18) vanishes, 
and we obtain an expression for E in terms of the tangential electric field alone, 
viz: 



(3.22) 



E(r) = - f dS' (n' X E(rO)-(V' X r^^^(r', r)) 



which exphcitly demonstrates the first part of the existence theorem that moti- 
vates this discussion. It may now be remarked that, inasmuch as (18) implies 
no particular boundary conditions on r, we may write the alternative expression 



E(r) = -ik f dS'(n' X H(rO)-r^'^(r', r) 



(3.23) 



- f dS' (n' X E(rO)-(V' X r^^^(r', r)), 

J S' 

which, when aS' coincides with S, reduces to 

(3.24) E(r) = -ik f dS' (n' X H(rO)-r^'^(r', r) 

•^ s 

the explicit formulation of the second statement of the fundamental existence 
theorem. 

The magnetic field can be calculated directly from the expression (19) for 
the electric field, or, more elegantly, by repeating the steps which led to (18), 
but replacing E by H and r^^^ by r^^\ It is evident that all that is necessary 
is to perform this substitution in the final formula, provided one also replaces 
H by — E (the minus sign is required to preserve the Maxwell equations). Hence 

H(r) = ik f dS' (n' X E(rO)-r^'^(r', r) 

- [ dS' (n' X H(rO)-(V' X r^^^(r', r)), 

J SI' 



(3.25) 

or equivalently. 



(3.26) 



H(r) = ik f r''\T, rO-(n' X E(rO) dS' 
Js' 

- V X [ r'''(r, rO-(n' X H(rO) dS'. 

J S' 



366 (S12) ELECTROMAGNETIC WAVES 

The magnetic field obtained from (19) is easily seen to agree with (26). If the 
surface S' coincides with S, 



(3.27) H(r) = ik f r''\T, rO-(n' X E(rO) dS' 

J s 



indicating that the field can be produced by the presence of suitable magnetic 
currents on the surface S. Replacing r^^^ by r^^^ in (25), we have the alternative 
expression 

H(r) = ik f dS' (n' X E(rO)-r^^\r', r) 
Js' 

(3.28) 

- [ dS' (n' X H(rO)-(V' X r^^^(r', r)), 

J S' 

which, when S' coincides with S, reduces to 

(3.29) H(r) = - f dS' (n' X H(rO)-(V' X r^^^(r', r)). 

J s 

Equations (27) and (29) are particular manifestations of the fundamental 
existence theorem. 

In concluding this section, we remark on the free space fields generated by 
an electric current on the surface of a perfect conductor. Since the current 
density is related to the tangential magnetic field at the surface b}^ (20), where 
the normal n' points into the conductor, we have 

(3.30) E(r) = -ik j dS' (n' X H(rO)-r^°^(r', r), 
and 

(3.31) H(r) = ^ f dS' (n' X H(rO)-(V' X r''\T', r)). 

For a plane current sheet, the current density is the difference in tangential 
component of the magnetic fields on opposite sides of the sheet. 

4. First Variational Principle 

With the stock of information concerning tensor Green's functions, we re- 
sume consideration of the diffraction problem. In this section, the development 
is based on the existence theorem as it relates to boundary values of the tangential 
electric field. Thus, in the half space bounded by the plane z = and an in- 
finitely remote surface where z is positive, we obtain from (3.22) and (3.27), 



HAROLD LEVINE AND JULIAN SCHWINGER (S13) 367 

E.(r) = f (e. X E(p'))-(V' X r':\x', y', 0, r)) dS' 

'^ St 

<4.1) , s > 

H.(r) = -ik f rl'\T, e').(e, X E(90) dS' 

where p denotes a position vector in the plane of the screen, and r+^*^^^ are 
the half space electric and magnetic field tensor Green's functions. The in- 
tegrals in (1) extend over the aperture only; on the remainder of the plane 
z = 0, the surface integrals vanish by virtue of the boundary condition (2.3) 
for the tangential electric field. Moreover, the surface at infinity does not 
contribute, as can be inferred from the known (radial) behavior of the electric 
field in the wave zone and the asymptotic properties of the Green's functions. 
On the other side of the screen, 

E_(r) = Eo(r) - f (e. X E(e'))-(V' X rL'\x', y' , 0, r)) dS' 

''St 

<4.2) z < 

H_(r) = Ho(r) + ik [ ri^^(r, pO'Ce. X E(eO) dS' 

'' Si 

in view of a change in the sense of the positive normal at the plane z = 0; Eq , 
Ho are given explicitly in (2.5), and 

r_(r, rO = r^(r - 2e,Te, , r' - 2e..r'e,). 

Combining (2.1) and (4.1) in accordance with (2.9) we obtain 

e^ X h' exp {zAn'-p} 

= -ike, X [ ri'\&, eO-(e. X EA9')) dS' 

•^ Si 

(4.3) 

= -2ike, X f r''\g, eO-(e. X E.,(eO) dS', g m S, 

as a vector integral equation to specify the tangential electric aperture field, 
the latter being explicitly linked with the propagation direction of the incident 
plane wave. Were the solution of (3) generally feasible, a single integration 
according to (1) or (2) would determine the free space fields at any point. To 
alleviate the practical difficulties of such a program, we shall devise an approxi- 
mation procedure for calculating the fields at large distance from the aperture. 
For the distant transmitted fields (1), we employ the asymptotic forms of 
the Green's functions (of. (3.12), (3.13)) 



368 (S14) ELECTROMAGNETIC WAVES 

^ J_ / exp {ik(r -n-rQ} -^ / _ 2e e ) exp {ikjr - n-^ - 2e,e,T0)} \ 
47r \^ r ^ z zj ^ J 



(4.4) 



1 Y exp {ikjr — n-rQ} exp {ikjr — n-(r' — 2e,e^-r0) 

- 7 2 V V I ' 

^irk \ r r 



■^), 



r 
n = - , r -^00 
r 



Thus, with elementary vector manipulation , 

E.(r) ^ £^ (e. X E„,(,')) X V- ^ [e exp W^^- n-r')! 

- (e - 2e,e,) expW>--n-y-2e.e..r'))! l ^^, 
(4.5) *■ ^ 



ik_e 

47r / ./s 



- (e, X En,(p)) X [n.£ - (n - 2e.e,-n) 



• (e — 2e,eJ] exp {— 2/bn-p} c?5. 
To simplify (5), we observe that, using the abbreviation V = e, X En' , 
V X n-£ - V X (n - 2e,e3-n)-(£ - 2e,e,) 

= 2e,(V X n-ej + 2(e,-n)V X e, = -2n X (e, X (V X ej) 

= -2n X V + 2(e,-V)n X e, = -2n X V, 
since e^'V = 0. Hence, 

(4.6) E+(r) ^ n X A(n, n') — , r -^oo 
where 

(4.7) A(n, nO = ^ / e, X E^^Cp) exp [-ihi-Q] dS. 
Further, 

H,(r) ^ _ I £ [e expia(r^-n-e01 + (^ _ ge^ej ex p W'" - n-pOl j 

• (e. X EAe')) dS' 



HAKOLD LEVINE AND JULIAN SCH WINGER (SI 5) 369 

, _L r ^^J exp likjr - n-rQI 
■^ 47r/c i^, ^ L r 

_ expWr-n.(r--2e.e.T0)n .^^^ ^ ^^^^^^^^ ^^, 

y Jz'=0 

= g^ /^ [nn.(e. X E.,(e)) - (e. X EAo))] exp {-ihi'^} dS, 



and thus 



6^*' 



(4.8) H+(r) c- n X (n X A(n, n')) — , r-^co. 

The results (6) and (8) show that electric and magnetic fields in the wave zone 
(kr ^ 1) are of equal magnitude and together with the propagation vector form 
a mutually orthogonal triad. 

Let us next perform scalar premultiphcation with En./(p) in the integral 
equation (3), and integrate over the aperture domain. There results 

[ e. X E^.,{g)'r''\g, pO-e. X EA9') dS dS' 

(4.9) = 2k^''J ^^ ^ ^-"(^) ^^P {ikn'-g} dS = ~W'k{-W, n") 

= ^h".£ e. X EAq) exp {t/cn-.p) dS = ~h" • k{-n" ,n') , 

making use of evident sj^mmetry in the first term as regards exchange of the 
indices n', n''. From (9), we learn that 

^ (h".£ e, X EA9) exp {-ikn"-Q] dS + h'. 1^ e. X E.^,,(g) 

(4.10) . exp {ika'-Q] dSj 

[ e. X EA&)-r''\Q, 90-e. X E_.,,(eO dSdS' 






= h"-A(n", n') = h'.A(-n', -n"), 

and this expression for h''-A(n", n'), or h'-A(— n', — n'O, is stationary with 
respect to independent variations of e, X Eq/ , e^ X E-^>> relative to tangential 
electric aperture fields specified by integral equations of the form (3). On 



370 (S16) ELECTROMAGNETIC WAVES 

carrying out individual scale transformations of the aperture fields in (10), we 
arrive at the useful stationary homogeneous forms 

' 1 =-4. 



h".A(n",nO h'.A(-n', -n") 
(4.11) 

* (h"-Js,e,XEn'(e) exp{-ibi''-Q}dS){h'-Js,e,XE.^.,{Q) exp{ikn 'g}dS) ' 

Since the integral equation (3) reveals that the projection of n' on the plane 
of the screen characterizes the aperture field e^ X En' , when excitation is re- 
stricted to the haK space z < 0, a reversal in the sign of n' implies an increase 
of IT in the azimuthal angle (p', the magnetic polarization vector being unchanged. 
However, as the aperture presents the same aspect in either half space, the 
field e^ X E_n' can be ascribed to plane wave excitation in the direction opposite 
to n', retaining the original magnetic polarization and reversing the electric 
polarization to ensure the correct sense of propagation. The equality of 
h''-A(n''', n') and h'-A(— n', — n'') is thus in the nature of a reciprocity relation 
for the diffracted amplitudes which accompany excitation and observation along 
a pair of directions in space. 

5. Second Variational Principle 

An independent variational principle stems from the existence theorem 
which relates to boundary values of the tangential magnetic field. For the 
half space ^ > with boundary values (z — 0) 

K\q) = e. X H,(9), 9 on S, 

(5.1) 

= e, XH^-^Xp), 9 ^ S, 

the last from (2.9), we have according to (3.24) and (3.29), 
E,(r) =ik f r':\T,g^)'KUgOdS^ 

+ ik f rl'\T, eO.(e. X hO.exp likn'-g'} dS' 
(5.2) 

H.(r) = I K:,(eO- V' X ri'\x', y\ 0, r) dS' 

+ f (e, X hO exp {tA:n'-e'l • V X t':\x' , y', 0, r) dS\ 

The function K^, multiplied by c/47r to obtain the equivalent surface current 
density, tends to zero with increasing distance from the aperture. 



HAROLD LEVINE AND JULIAN SCHWINGER (S17) 371 

Likewise, the boundary distribution 

K-(9) =e, X H_(e), g on S, 

(5.3) 

= e, X H^'^Xp), e in 5, 

for the half space z < 0, yields 

E_(r) = e' exp {^An'-r} + e''(e — 2e,e,) exp {^/bn'•(r — 2e,e,-r)} 



-ik f ri^^(r, eO-K-,(90 

- ik [ r'^ir, pO-(e, X h') exp {z/bn'-p'} dS', 



(5.4) 

H_(r) = h' exp {ikn'-r} - h'-(e - 2e,eJ exp {ika'-(i - 2e,e,-r)l 

- f K-,(90- V X r'y(x% y\ 0, r) dS^ 

- f (e, X hO exp {ika'-g^} • V X ri^^(a;', y', 0, r) rf>S' 

where the integrated terms are due to the infinitely remote part of the boundary 
surface. At large distance from the aperture, K~ (apart from the factor c/4:ir) 
becomes identical with the infinite screen current density, 

(5.5) Koio) = e. X Ho(9) = 2e. X ^'^"(9), 9 on S, + S, , 

From the requirement of vanishing tangential electric field on the shadow 
face of the screen, it follows that 



e, X j r^^^fo, o')'K:.(o') dS 
(5.6) 



.X f r^°^(9, 90-k:,(9') 

'' St 

= -e, X f r''\g, 90- (e, X hO exp lika'-g'] dS', 9 on S^ 

'' Si 

since only tangential components of r+^ are involved. Similarly, on the 
illuminated face of the screen, 



(5.7) 



e, Xe' exp {ikn'-g] = ike, X [ r^'^g, Q')'K-,{g') 

'J s. 



dS' 



+ ike, X f r''\Q, 9')-(e, X h') exp {ikn'-g'} dS', q on S2 . 

'' S^ 



372 (S18) ELECTROMAGNETIC WAVES 

The integral equations (6) and (7) for Kn' and K"- , like that encountered in 
the previous section, have no ready solution. It may be noted that for a com- 
pletely infinite screen, equation (7) becomes 

e, Xe' exp [Hm'-Q] = 2ike, X [ r''\g, p') 

^ Si + S, 

(5.8) 

•(e, X hO exp {ika'-Q'} dS', 9 on ,Si + S2 

a result which can be directly verified by use of (3.12) and the integral repre- 
sentation 



exp {ik I r — r^ I j ^_ r 

47r I r - r' I ~ St j_< 



dk^ dk. 



(k' -ki- kir' 

(5.9) 

. exp {iihix - x') + Kiy - y') + {k' - kl - klf'' \z-z' \)]. 

Using (4.4) we deduce that the transmitted fields remote from the aperture 
are 

E+(r) ^ -n X (n X B(n, nO) — 
(5.10) r-^00 

H,(r)-nXB(n,nO'-J^ 



where 



^(->-')=l 



(5.11) 



f Kn^(9) exp {-ikn-g} dS 

+ (e, X hO j exp {ikin' -nj-g] dSj. 
In consequence of the integral equation for K^' , 

= - [ dS' f rfiSK:,^(e)-r^°^(9, 90-(e. XhO exp {zAn'-e'} 

= - f dS' f dSKUQ)'r''\g,g')'(e,Xh'')explikn^^'9' 
and furthermore, by appealing to (6) and (8), 



HAROLD LEVINE AND JULIAN SCHWINGER (S19) 373 

[ dS^ f dSK:.,(g)-r''\g, 90- (e. X hO exp {ikn'-Q'} 
= [ dS' f dS K:.,{Q)-r''\g, 90-(e. X hO exp {ikn'-g'] 

- f dS' dS K:,,{o)-r''\g, 90-(e, X hO exp {ikn'-Q'} 

•^ Sa 

= f dS' f dS Kn^^(9)-r^'^^(e, 90-(e. X h') exp {^A3l'•9^ 
+ [ dS' [ dS (e. X h^O exp {ikn'^'g} 'r''\g, Q')-(e, X h') 

JSi + S, J s, 

• exp {iJm'-Q'] 

- f dS' dS{e, X h'O exp {ikn'' -gir'^'ig, 90-(e, X h') exp {ikn'-g'} 

J Si 

= ^^'-[j^^ k:„(9) exp {iJm^-Q} dS 
+ j^ (e, X h'O exp {^/c(n' + n'0-9} ^^^J 

- [ dSdS' (e, X h') exp {iJm' - g} -r'^'ig, gO-(e, X h") exp {ikn''-Q'], 

^ Si 

Thus 

f KUg)-r''\g,gO''K^-n"i9')dSdS^ 

+ f dS' f dSKUg)'r''\g,g^)-{e,Xh'')exp{-ibi'''g'] 

'^ Si -^ s, 

(5.12) ^ f dS' f dSKl^,,{g)-r''\g, g^)'{e, XW) exp {ikn''9'} 

•J Si J Sa 



= -pe"-[B(n",n') - B^(n", n')] 



= - pe'-[B(-n', -n") - B^(-n', -n")], 



374 (S20) ELECTROMAGNETIC WAVES 

where 

e".B^(n",nO =e'.B^(-n', -n") 

(5.13) = -- f (e. X hO exp {iJai^ - q} -r'^'ig, 90-(e. X h") 

. exp {-^A3l"•9'i dS dS\ 

In accord with considerations of the preceding section, the function Klj,' 
arises from plane wave excitation along the direction obtained by rotating n' 
about the 2;-axis through an angle w. If K"^ vanishes on the screen, the ampH- 
tudes B, Bk are equal, and hence the latter befits a Kirchhoff approximation. 
It may be noted that the real part of B^ is a divergent integral, although 
compensating integrals of (12) render B entirely convergent. The expression 
(12) for e''-B(n'', n') or e'-B( — n', — n'O has the property that it is stationary 
with respect to independent variations of Kn' , K^n" relative to solutions of the 
integral equation (6). 

The foregoing development can be modified slightly, so as to involve the 
screen current density 

(5.14) K(q) = K-(9) - K"(e), e on S, 

instead of K^, K~ individually; this transformation reflects a change in geo- 
metrical basis, with the screen now regarded as an obstacle imbedded in free 
space. Following the discussion at the close of section III, and ^-ith particular 
reference to (3.30) and (3.31), the fields at any point are 

E(r) = e' exp {^A:n'•^) - ik f KA9')'r''\&', r) dS' 
(5.15) 

H(r) = h' exp {ika' -r} - f K^q')'^' X r''\x', y\ 0, r) dS' , 

and clearly, from (15) or (6) and (7), the integral equation to determine K is 

(5.16) e, Xe' exp [ihi'-Q] = ike, X f r^°^(p, q')-KA9') dS', q on S2 . 

If the current density K is resolved into two parts, 

(5.17) K(p) = K(9) + Ko(9), 9 on S, 

with Ko given by (5), and hence K a null function for 9 ^^<^, it can be sho^^Ti 
that 



HAROLD LEVINE AND JULIAN SCH WINGER (S21) 375 

(5.18) {^^Jj} = {g^^J + 2ik j^ r^°^(r, 90-(e. X h') exp {tAn'-p') dS^ 

and 

{nlr)} = {h^)} + 2 /.. ^^' ^ ^'^ ^^ {ika'-Q'\-W' X r'°'(^', y', 0, r) dS' 

(5.19) . _ 

K..(e')- V X r'°'(x', y', 0, r) dS', 

whence 

e. X f r'°'(e, e')-K„.(e')dS' 
(5.20) 

= 2e. X f r^°^(9, pO-(e. X h') exp {zAn'-?') ^>S', p on ^S^ . 

An inspection of (6) and (20) reveals 

(5.21) K(e) = -2K^(e) 

as can be inferred also from the general considerations of^ section 2. It is evident 
that to obtain a stationary expression for B in terms of K, one need only perform 
the substitution (21) in (11) and (12). The Kirchoff approximation Bk is thus 
based on the induced current Kq , unmodified by the presence of an aperture. 

6. Transmission Cross Section 

To extend the practicality of the variational principles, we develop their 
connection with a quantity of physical interest, the plane wave transmission 
cross section of the aperture. A suitable expression for the cross section, or 
ratio of transmitted energy flux to incident energy flux per unit area, is derived 
from the real part of the Poynting vector theorem in a non-dissipative source 
free region, (the asterisk denotes complex conjugate) 

(6.1) (Re I V • 1^ E X H* c?r = 
which involves the quantity 

(6.2) S = ^EXH* 

OTT 



376 (S22) ELECTROMAGNETIC WAVES 

the real part of which is the average energy flux vector. On integrating (1) 
throughout the shadow half space, we establish a connection between the total 
energy flux at infinity and that through the aperture, 

(Re-f [ n-E+ X m dS 
(6.3) 

= (Ref- f e,-E X K* dS = (Re ^ [ e, X E.(H^^^)* dS, 

OTT J Si OX J Si 

taking account of (2.3) and (2.9). Since the magnitude of (2) for the incident 
wave (2.1) is c/Sir, the cross section becomes 

(6.4) (7(nO = (P^eW- [ e, X E^'ig) exp {-ilcn'-o] dS = ^ ^m h'-A(n', n'), 

and in the latter form, the stationarj^ expression (4.11) for h'-A(n', n') is directh^ 
applicable. 

It can be verified that 

(6.5) (Re f e, X E.(H^^'=)* dS = -(Re [ e. X H.(E^^<=)* dS, 

-J Si 'JSi + St 

which proA'ides another form of the cross section 



(7(nO = -(Ree'- e, X Hn'(p) exp {-I'An'-o} dS 

(6.6) 

= - % £rm e' -6(11', nO, 

A/ 

adapted to the stationary principle of (5.12). The same result is established 
also via the sequence of relations 

ll'•A(n^nO = -e'-n' X A(n', nO =e'-n' X (n' X B(n', n')) = -e'-B(n',nO, 

since e'-n' = 0. 

Combining (4) and (6) with (4.11) and (5.12) respectively, we find 

<r(n) = - 2^^^ L^'*'^''^ ^' ^ ^""^^^ ^^ {-ikn-o} dS 

.(6.7) 

jh' i s. e, X E_n(p) exp {iJm-o] dS) 

7 J 



js. e. X E,(9).r^°^(9, 90 -e. X E_,(9') dS dS' 



and 



HAEOLD LEVINE AND JULIAN SCHWINGEB (S23) 377 

<T(n) = <r^(n) - 2k Sm I /«. dS' f^. dS Kis) ■ r'°'(e, e') -(e. X h) exp { -ika-g'] 



(6.8) 

/.. dS' / 

Ss.K:{s)-r""{g, e')-KUe') dS dS' 



h, dS' !s, dS K!„(e)-r'°'(e, eO-(e, X h) exp {ika-e'] '\ 



where 

(7K{n) = 2k^m [ (e, X h) exp {ikn-g} -t'^^q, pO'Ce. X h) 

(6.9) -^'^ 

• exp {-ikn-g'} dSdS' 

is the Kirchhoff contribution. In (7)-(9) we have a pair of stationary, homo- 
geneous expressions suitable for evaluating the cross section. A comparison of 
their respective predictions, based on trial functions for E and K^, provides an 
estimate of the accuracy obtained, since the results are identical only if the 
correct functions are employed. Particular interest concerns features of the 
cross section at very long or short wave lengths compared with aperture dimen- 
sions; such details as frequency or angle dependence are available from the 
stationary principles without knowledge of the correct boundary field distribu- 
tions, which merely fix the proper scale. 

At long wave lengths, or low frequencies, the characteristics of diffraction 
by an aperture can be described quite generally, without restriction to a partic- 
ular type of incident field. The secondary field is then attributed to a pair of 
electric and magnetic dipoles in the aperture. These dipoles are respectively 
normal and parallel to the plane of the aperture, and are related to the corre- 
sponding components of the fields Eq , Hg , which can be regarded as constants. 
The coefficients of proportionality, termed electric and magnetic polarizabilities, 
are independent of wave length and given in terms of aperture dimensions. For 
the electric dipole, a single polarizability occurs, whereas the magnetic dipole 
has an associated polarizability dyadic, as may be inferred from the relative 
orientations of moments and exciting fields. Symmetry of the dyadic imphes 
that there are mutually perpendicular principal axes, along each of which the 
magnetic dipole moment is a multiple of the associated component of Hq . 

The tangential electric aperture field has separate contributions due to 
Eo , Ho , and for plane wave excitation takes the form (cf . Appendix 2) 

(6.10) E,(p) = e,-eV0i(e) + iki-h-lmcf^^iQ) + h-mlcl>,{g)), g m S^ 

where 1, m are unit vectors along the principal axes of the magnetic polarizability 
dyadic, viz.: 

(6.11) e, X 1 = m, e, X m = -1, 1-m = 0. 

The functions 0(p) are individually real and frequency independent; a boundary 



378 (S24) ELECTROMAGNETIC WAVES 

condition is necessary to ensure that the tangential component of £« vanishes 
at the rim of the screen, where (f>i{g) itself is zero. 

If the electric field of an incident plane wave is perpendicular to the plane of 
incidence defined by e^ and n, whence e^-e = 0, we find on use of (10), (11) 
and the expansion 

(6.12) r<»'(r,r')=-jJpVV'|:^ + i|+..., /. ^ 

that the cross section (7) becomes 

, , . 47rA;' 
.(n) = — 

(6.13) 

. (/..{h.ll(/„(e)+h»min<^3(e)l^^'(J.J(h>y0.(e)+(h-m)^03(o)}^>S)^ 

(/5,{h-ll02(e)+h-mm03(p)!-VV'r--^-{h-ll02(eO+h-mm93(pO}^5J^^^ 

k -> 0, e, -e = 

The latter expression reveals a X~* wave length dependence, characteristic of 
Rayleigh scattering. A simplification of the rather involved angular dependence 
occurs, for example, in the case of a circular aperture, where 02(0) = ^3(9) = 
4)(p), and the principal axes coincide vdth any pair of perpendicular radii; it 
turns out that 

, . ^ IM! (/., <f>(p) d^' 

""^^ ~ 3 (/., Midi + dl) [1/(1 g - g' I)] 0(p') dS dSy ' 
(6.14) 

k-.0, ere = «- q < 9 < 2x. 

The cross section (14) may be compared with a corresponding result [8. I, eq. 3.7^ 
in the problem of scalar diffraction theory, where the wave function vanishes at 
the screen, although the aperture has any shape; we find that for a circular 
aperture, the electromagnetic cross section is the larger by a factor of S. 

With arbitrary incident plane wave polarization, the cross section has a 
more complicated angle dependence, although the factor X~^ is retained. It is 
simpler to obtain the angular features via the diffracted intensity, 

(6.15) (7(n) = f ' d^' f ^ dct>' sint?' | n' X A(n', n) |', 

once the amplitude A is derived from the tangential electric aperture field. 

At very short wave lengths, or high frequencies, the problem again simph- 
fies, based on the nearly geometrical nature of wave propagation. This char- 
acteristic finds expression in the trial functions for (7), 



HAROLD LEVINE AND JULIAN SCHWINGER (S25) 379 

(6.16) e, X E.„(9) = e, X e^M exp {^ikn-g} 

which differ from the incident fields only by a real modulation factor $0(9), 
assumed to be frequency independent or of limited variation in distances com- 
parable to the wave length. It follows that 

o-(n) = — Ok ^^^^ Aj ^M d^j '^^ J ^z X 6*^(9) exp {ika-g} 

(6.17) '^ '^ 

. r''\Q, 90-e. Xe$,(90 exp {-ika-g'} dS dS' j 

noting 

h-e, Xe = h-e, X (h X n) = ne, = cos??. 

A reduction of the multiple integral in (17), with asymptotic validity for A; —» 00^ 
is possible on account of the attendant rapid exponential oscillations. This 
feature suggests extension of the integration domain for the primed coordinates 
(say) to the entire plane Si + S2 , and identification of arguments for the $n 
functions, the last since oscillations of the integrand are least rapid if 9', q are 
close together. Hence 

[ e, Xe^n(9) exp {ika • q} -r'^'ig, 90-e, X e<l>n(90 exp {-ika-g'} dS dS' 
c^ j dSe^X e^lig) exp {ika-g} -{t + p Vv) 

Jsr + s, 47r I 9 — 9 I 
• exp { —ika-g] -(e, X e) dS 



2k cos 



- [ $^(9)e, X (h X n) •(£ - nn) -e, X (h X n) dS 

V J Si 



= ^cos??| ^1(9) dS, 



and thus 



380 (S26) ELECTROMAGNETIC WAVES 

A stationary value of (18) is attained with $n = const., and the resulting geo- 
metrical cross section equals the projected area of the aperture on a plane 
normal to the direction of the incident wave. 

From the Kirchhoff cross section (9), we find 

(6.19) <r^(n) = ^^ (e. X h)^ k^O 

the wave length and angle dependence at variance vdth pre\nous results. How- 
ever, for short wave lengths, arguments similar to those used in the derivation 
of (18) yield 

(6.20) o-i,(n) ^ Si cos ??, k-^oo 

in accord with the stationary value of (18); the second term of (8) vanishes in 
this limit, independently of K^, as revealed by the mutually exclusive integra- 
tion domains for the numerator. A long wave length approximation for the 
latter term, adequate to correct the Kirchhoff deficiencies, requires a precise de- 
scription of K"^, and is therefore compHcated. 

7. Diffraction by a Circular Aperture 

To illustrate the variational techniques, we consider the diffraction of plane 
waves by a circular aperture, with the restriction of normal incidence. B\^ 
geometrical symmetry, the tangential electric aperture field has a component 
along the incident electric polarization direction only, and is a function of the 
radial coordinate. Hence, if A = A(0, 0), we learn from (4.11) that 

(7.1) A, = A.= -1- (!s,<^(.)dSr 



2^^ A. 0(p)[rf.'(e, e') + ni'ie. e')]*(p') dSdS' 

An appropriate expansion for 0(p) is 

TO / 2\n-l/2 

(7.2) 0(p) = E^nU -^) 

n=l \ "■ / 

where a is the aperture radius, and the Ar, are arbitrary coefficients; the leading 
term of (2) has a form predicted by the low frequencj^ integral equation. De- 
noting 

(7.3) B„ = /Jl - g"-'%. = ^-^ 



271+1' 



and 



// 2\n!-l/2 / /2\n-l/2 



HAROLD LEVINE AND JULIAN SCHWINGER (S27) 381 

we get 

(7.5) A^ f: Ar^Ar^Cmn = -^{jl A^b)\ 

The result of differentiating with respect to ^^(say), and invoking the stationary 
property of ^^ yields, after some manipulation 

(7.6) A^= -^i: B^D^ 
where 

(7.7) f:C^nD^ = B^, m= 1, ... . 

n=l 

If the linear equations (7) are reduced to a finite set with the same number A^ 
of unknowns, an approximate value Ai^^ is deduced from (6). It can be shown 
that 

where £)„ is the determinant 1 1 (7^„ 1 1 of n rows and columns, and 3D„ is obtained 
from the latter on replacing the last column hy B^ , - - - B„ . 
Employing the integral representation 

exp {ik I p - pMi 
47r I p - eM 
(7.9) 



^irf '^'^^^^^ + ^'' ~ ^^^' "^^^ ^^ ~ ^'^^' 



)7T2 



r^r 



(f - k') 



2\l/2 J 



arg (f^ - ky = 0, t>k; = -^2, f < k 
and the Bessel function addition theorem, 

Joiiip' + p" - 2pp' cos (0 - 0'))"') 
(7.10) 



E (2 - 5„„)/,(rp)/„(fp') cosn(0 - 00, S„ = {' ^ "^ * 

1, p = 9 



we find 



vlVie, e') + rir(e, e') = j- 2 (2 - W cos «(0 - 0') f " r (^f 

47r Jo 



382 (S28) ELECTROMAGNETIC WAVES 

whence 

r" / 2\m-l/2 /.o / /2\n-l/2 

■ i. 'I' - ?) «« * i. »V - &-) «?/) <i/ 

=i(r>K»+i)K«+i) 

• f " [(«'' - 1)"" - {v'' - \y"]v-"'*'"J„^Uk(m)J.,Uhca,) dv. 
The two integrals of (11) are simply related, for if 



(7.12) 



then 



(7.13) 



m + n-2 I f 2 2\l/2 ~f,m + n) t f \ T / \ J 

= a \ {v - a) V Jm+i/2(v)Jn+i/2(v) dv, 

Jo 

= «"*" r {v' - o?y'"v-'-*"j^,Uv)J.»,M dv 

Jo 



= (m + n - 2)FUoc) - aFUc^, 
where the prime signifies differentiation with respect to the argument. Thus, 

(7.14) 

' [{m + n - 3)FUka) - kaFUka)]- 

The function Fmniot) has been considered elsewhere [8, I, 4.16 et seq.], ^^dth 
its real and imaginary parts given explicitly f or m, n = 1, 2. Having this in- 
formation, we return to equation (8) and obtain the transmission coefficient 
(transmission cross section/area of aperture) 



t ^ 2 ^^ 7 2 5W -^x 

ira ka 



(7.15) 






HAROLD LEVINE AND JULIAN SCHWINGER 



(S29) 383 



where 
(7.16) 



,(1) 8 



r ' = —ka^m 



dw F,,{ka) + kaFUkd) * 



(7.17) 






^m 



' F22{ka) - kaFUka) - {l/2b){kaf{F,,{ka) + kaF^ka) - lOFUka)] 
{FUka) + kaFUkd)]{F22ikd) - kaFUka)] + {kaFUka)}^ 



€tc. Curves illustrating the variation of t''^^ and t''^^ ior < ka < 9 are pre- 
sented in Figure 2. An expansion in powers of ka gives 



(7.18) 



''" ~ 27,r^ ^^ 



«){ 



1 + II (te)' + 0.72955(te)' + • 



] 




I t I 1 I t I I 1 t » 



Fig. 2. Transmission coefficient of circular aperture for normal incidence of plane electromag- 
netic waves, a = radius of aperture, k = 27r/(wave length). i^^\ first variational approxima- 
tion, based on tangential electric aperture field of the form (1 — (pVa^))^'^- i^'^\ second 
variational approximation, based on tangential electric aperture field of the form Ai(l — 
{p^/a^)yi^ + ^2(1 — (p^/a^))^!^. Ik , Kirchhoff approximation, using Stratton-Chu formulation. 

tn , Rayleigh-Bethe approximation. 



which confirms the rapid initial rise of transmission coefficient as the parameter 
increases from very small values; this feature makes it evident that the leading 



384 (S30) ELECTROMAGNETIC WAVES 

term of (18), or Rayleigh-Bethe result, is accurate only at extremely long wave 
lengths. Furthermore, 

(7.19) e' = ^ {kay\l + II {kaf + 0.74155(A:a)^ +•••], 

which differs from (18) in the term of relative order {kaY. Each approximation 
t'^^^ gives correctly the numerical factors for powers of ka less than the square 
of that by which the corresponding electric aperture field is in error, both relative 
to the lowest powers; in particular, t'^^^ is exact through terms of power {kaY* 

In the short wave length limit, the transmission coefficient is conveniently 
obtained from (6.18), with the result 

(7.20) '"' ^ 1 - (2FTly ' ^"^"^ 

as for the scalar diffraction problem of [8, I]. 

It is of interest to compare the variational predictions with those yielded hj 
Kirchhoff approximations. The Kirchhoff amplitude AK(n, n') which stems from 
(4.7) on identification of aperture and incident electric fields is 

A;,(n, nO = ^ / e. X E;r(p) exp {-ikn-Q} dS 
(7.21) 

= f- (e. X eO f exp lik(n' -n)-g} dS; 

thus, for electric polarization along the a:-axis and a circular domain Si , 

Ax(n, 0) = e^l 27r v esc t}Ji{ka sin ??) ), 

so that, using (6.15), 

^i. = 1 - ^7- JM dt 
(7.22) ^'"'^ '^' 

= {kd)ys, ka-^0; o^ 1, ka -^co . 

Furthermore, the Kirchhoff transmission coefficient obtained from (6.9), based 
on the infinite screen current distribution, agrees completely ^^^th (22). Using 



*J. Meixner and W. Andrejewski, Annalen der Physik, Volume 7, 157 (1950) have derived 
the result 

t = (64:/277r^)(kani + 22/25{kay + •••] 
from a calculation using spheroidal functions. 



HAROLD LEVINE AND JULIAN SCHWINGER (S31) 385 

the Stratton-Chu formulation, which involves the incident electric and magnetic 
fields in the aperture and on its rim, it turns out that 

1 1 r2^<i 

t^= l-\ (Jl{ka) + JKka)) - -^ Ut) dt 

(7.23) ^ ^"'^ -^^ 

= {7/24){kay, /ba -> 0; ^ 1, ka -^co 

which differs slightly from (22) . According to the numerical results, all Kirchhoff 
transmission coefficients fail to account for the actual resonance behavior at 
wave lengths comparable to the aperture dimension, and moreover give a wrong 
order of magnitude at longer wave lengths. It is therefore evident that the 
variational formulations enjoy considerable advantage for the practical analysis 
of diffraction problems. 



Appendix I 

The Tensor Green s Function of Infinite Empty Space 

The free space tensor Green's function is defined as a solution of 

(A.l) V X (V X r^"^(r, rO) - k'r''\T, r') = ^8{T - r') 

upon which are imposed the requirements that all of its components vanish at 
infinity, and that it satisfy the radiation condition. Its construction is facilitated 
by first evaluating the divergence of the differential equation it obeys, which 
yields 

(A.2) /bV-r^"^(r, rO = -V6(r - r') = V'5(r - r'), 

and then employing the vector identity 

(A.3) V X (V X ) = V(V- ) - V'( ), 

to obtain 

(A.4) (V' + k')T''\r, rO = -(e - J, V V')5(r - r'). 

The latter equation can be satisfied by writing 

(A.5) r""(r, r') = (e - p VV')(?(r, r'), 



386 (S32) ELECTROMAGNETIC WAVES 

provided the scalar function G obeys 

(A.6) (V^ + k')G{r:, rO = - 5(r - r'). 

The divergence equation (2) imposes an essential restriction on G, for 

(A.7) k'V'T''\x, r') = V'6(r - r') + ^'(V + V')G{r, r'), 

and therefore 

(A.8) ^(V + VO(?(r, rO = 0, 

which states that G{x, x') must be a function of r — x'\ the well known solution 
of the differential equation for G, 

(A.9) G(r,rO^ ^"Pj^|jL-'i'l' 

is indeed a function only of the distance between the two points r and x'. The 
choice of sign in the preceding exponential is in deference to the radiation condi- 
tion, which requires spherical waves moving outward from the source at x'. 
Hence 



(A.10) r'>,.) = (e-lvv')^^^^Mi^ 



II 



is the tensor Green's function of infinite space, for it satisfies the differential 
equation, the radiation condition, and evidently all of its components vanish at 
infinity. 



Appendix II 

Low Frequency Aperture Electric Field 

To study the aperture electric field in the low frequency approximation, 
we start from the integral equation (4.3), written as 

e. X Ho„(e) = -^ik [ e. X t''\q, pO'Ce. X E^CpO) dS' 
(A.ll) 9 in ^1 

= -4^/c f e. X t''\q, q') Xe.-EJp') dS\ 

where the subscript t denotes tangential (or transverse to z) character. In- 
troducing the dyadic r^°^ explicitly, the latter equation becomes 



HAROLD LEVINE AND JULIAN SCHWINGER (S33) 387 

e. X nM = ^ / E,„(p') --vUk\,-,'\] ^g, 
T Js, \ e — 9 \ 

(A.12) 

- 4: f e. X V,V, ^-P\ik\9-9'\] xe..E,„(,') dS', 

and by use of the vector identity 

(A.13) e, X V,V, Xe, = V, X (Vi X ) = V,V, - e,V? , 

it follows that 

(V, X V, X -k^) f E.,(,')'-^SJM9SZf^as' 

(A.14) 

= iTrke, X Hon(e) 

Q in Si . 
= 2iTrke, X h exp {iAn-p} 

At low frequencies, the aperture electric field has a twofold composition, of 
magnetic (Ho) and electric (Eq) origin, respectively. The magnetic part is 
determined by the integro-differential equation (14) on setting k = everywhere 
in its left hand side and in Hon on the right hand side; the electric part is given 
by a corresponding alteration of the equation which results on taking the 
divergence of (14), namely 

V,. f E,„(e') 2^3LiMi^^ ^sv 

(A. 15) ^'^^ v7 V TJ ^ ^ 

= ■^®-* V X Hon(9) 

= '7re,-Eon(p) = 2x6, -e exp likn-g]. 
Consequently, the basic low frequency equations are 

(A.16) V, X ( V, X f E,(eO T—^^^j = 2Tike, X h 

and k —^ Oj q m Si 

(A.17) Vr f E,(eO I "^^^ , = 27re.-e 

^sx I 9 - 9 I 

omitting a subscript which refers to the incident plane wave propagation direction; 
A further investigation reveals that the entire aperture field has a similar 



388 (S34) ELECTROMAGNETIC WAVES 

decomposition, with a magnetic part in which the normal magnetic field pre- 
dominates, and an electric part with predominant tangential electric field. For 
each type of excitation, there is an associated dipole moment, magnetic in the 
plane of the aperture and electric normal to the latter. According to the usual 
formulas, the magnetic dipole moment is proportional to the integral of e^ X 
E(p), or magnetic current density, over the aperture, while for the electric dipole 
moment the integral of (e^ X £(9)) X 9 is involved. 

As regards the form of the aperture field, we note that its tangential electric 
component may be written 

(A.18) E^(9) =e.-eV<^i(e) 

where </>! is a real, frequency independent function; it is readih' verified that 
(18) makes no contribution to (16), provided 0i vanishes on the aperture rim. 
For the tangential magnetic component, we have 

(A.19) Eh(9) = ik{-h'lm<l>,(Q) + h-mlcf^^ig)) 

where 1, m are unit vectors along the principal axes of the dj'adic which relates 
the magnetic dipole moment to Hq , and 

(A.20) e, X 1 = m, e, X m = -1, 1-m = 0; 

the functions 02 , 0.3 are real and frequency independent. To justify (19), 
observe that the related magnetic moment has components along the principal 
axes which are proportional to those of Ho . The total field Etig) = E^(o) + 
EH(e) thus has the form indicated by (6.11). 

In particular, for a circular aperture, 1 = e^ , m = e^ and 

1 ■ 1 ■ ^ /2 2x1/2 

- - {a - p) . 



2"^ 2 



'3 



Addition in Proofs: 



The variational results for diffraction by a circular aperture obtained in 
section 7 require important qualification. This is a consequence of recent in- 
vestigations by Bouwkamp (to appear in Philips Research Reports), which show 
that the low frequency aperture electric field for normal incidence has com- 
ponents both parallel and perpendicular to the incident electric polarization 
direction, and exhibits angular asymmetry. Specificallj^-, if the incident electric 
polarization is along the x direction, the aperture field components according to 
Bouwkamp are (omitting scale factors) 

f^^ J. 2a' - x^ - 2y^ xy in -^ 

^^^ ^' ~ {a' - x^ - yy ' ^^ ~ (a' - x' - y^ ' 



HAROLD LEVINE AND JULIAN SCHWINGER (S35) 389 

or in polar coordinates, 

(2a^ — p^) cos <p 



(B) 



^, = '-",. '".:ur , E, = -2(a^ - pT' sin v 



(a - p) 
With this field, the variational transmission coefficient turns out to be 



(C) 
where 



^ 16 Pika) 

-K P\ka) + Q\ka) 



9 9 



Q(.) = ^ /o(2.) + ^ /,(2.) + (l - ^ - ^) £" J,(0 dt, 

1.8 





o 




















/ 


\ 


















/ 








































1 

















^ 


























o 


MEIXNER AND ANDREJEWSKI 








/ 




















7 





















1.0 



2.0 3.0 



40 



ka 



5.0 



6.0 7.0 



8.0 



90 



10 



Fig. 3. Variational approximation to the transmission coefficient of a circular aperture, based 
on Bouwkamp's form of electric aperture field. Points computed from rigorous theory using 

spheroidal functions. 



390 (S36) ELECTROMAGNETIC WAVES 

and Jo , Ji denote the Bessel functions of order zero and one, while So , Si 
denote the corresponding Struve functions. 

An expansion of t for small values of ka yields 



^ ~ 277r^ 



(kayll + II (kay + 0.4079 W + • • • J 



which may be compared with the exact result of Bouwkamp (see also Meixner 
and Andrejewski), 



64 

277r^ 



(kayll + II {kaf + 0.3979M* +•••[. 



From agreement of the latter expansions through terms of relative order (kay, 
the correctness of the low frequency aperture field (A) is confirmed. The re- 
marks concerning accuracy of the variational approximations (7.16) and (7.17) 
in the text are therefore erroneous. 

At very high frequencies, ka ^ 1, the transmission coefficient in (C) ap- 
proaches zero, since P(ka) increases logarithmically while Q(ka) tends to unity. 
A null transmission coefficient can also be inferred from (6.18), for integrals 
of El and El over the aperture are infinite, in consequence of the field singu- 
larities at the rim of the screen. The latter feature of the low frequency field 
renders it a poor variational trial function when the frequency becomes very 
great, as the correct field is then more nearly constant over the aperture (for 
normal incidence); hence the corresponding variational predictions are inaccu- 
rate. From data of the accompanying figure, it can be inferred that the trans- 
mission coefficient (C) decreases slowly when ka ^ 1, where the alternative 
formulation based on screen current becomes appropriate. 



Bibliography 

1. J. A. Stratton and L. J. Chu, Diffraction theory of electromagnetic waves, The Physical Review, 

Volume 56, 1939, p. 99. 

2. Lord Rayleigh, On the incidence of aerial and electric waves on small obstacles in the form of 

ellipsoids or elliptic cylinders, on the passage of electric waves through a circular aperture 
in a conducting screen, The Philosophical Magazine, Volume 44, 1897, p. 28. 

3. H. Bateman, The Mathematical Analysis of Electrical and Optical Wave Motion, Cambridge, 

1915, p. 90. 

4. H. A. Bethe, Theory of diffraction hy small holes, The Physical Review, Volume 66, 1944, 

p. 163. Also E. T, Copson, An integral equation method of solving plain diffraction 
problem. Proceedings of the Royal Society (Series A), Volume 186, 1946, p. 100; and 
W. R. Smythe, The double current sheet in diffraction. The Physical Re\dew, Volume 
72, 1947, p. 1066. 

5. Joseph Meixner, Strenge Theorie der Beugung elektromagnetischer Wellen an der vollkommen 

leitenden Kreisscheibe, Zeitschrift fiir Naturforschung, Volume 3A, 1948, p. 506. 
0. P. Debye, Der Lichtdruck auf Kugeln von Beliebigen Material, Annalen der Physik (Series 4), 
Volume 30, 1909, p. 57. 



HAROLD LEVINE AND JULIAN SCHWINGER (S37) 391 

7. C. J. Bouwkamp, A note on singularities occurring at sharp edges in electromagnetic diffraction 

theory, Physica, Volume 12, 1946, p. 467. 

8. H. Levine and J. Schwinger, On the theory of diffraction by an aperture in an infinite plane 

screen, Part I, Physical Review, Volume 74, 1948, p. 958. Part II, The Physical 
Review, Volume 75, 1949, p. 1423. 

9. J. W. Miles, On the diffraction of an electromagnetic wave through a plane screen, Journal of 

Applied Physics, Volume 20, 1949, p. 760. Errata, Journal of Applied Physics, 
Volume 21, 1950, p. 468. Miles has also considered variational aspects of the electro- 
magnetic diffraction problem. The power transmitted through the aperture and a 
related aperture impedance are presented in stationary forms; however, stationary 
properties of the diffracted field amplitudes are not discussed. 



t 



On Systems of Linear Equations in the 
Theory of Guided Waves 



By W. MAGNUS and F. OBERHETTINGER 
California Institute of Technology 



Introduction and Summary 

We investigate the diffraction of an electromagnetic wave between two 
parallel planes or in a wave guide of rectangular cross section by a plane strip. 
We assume that all components of the field depend on the time t in a purely 
periodic way which can be described by a factor exp {io)t}, and that the frequency 
ca and the proportions of the wave guide are such that there exists essentially 
only one type of waves which is not attenuated. If the incoming wave is of 
the type exp {tax} cos I3y where a, (3 are real, the components of the diffracted 
wave can be expanded in a Fourier series. It becomes evident that the co- 
efficients of the series are uniquely determined by the condition of the finiteness 
of the total energy in any finite part of the space. This condition has already 
been used by Bouwkamp [1], Maue [6] and Meixner [7] who also showed that the 
components of the electromagnetic field behave at the edge of a plane diffracting 
obstacle like p"^^, where n = —1, 0, 1, • • • and where p is the distance from the 
edge. The Fourier coefficients are determined by an infinite system of linear 
equations; for a certain closely related system the existence of a solution was 
proved recently by Schaefke [11]. If the width of the diffracting strip is exactly 
one half of the width of the wave guide, the system of linear equations can be 
dealt with by a process of successive approximation, such that the first steps 
can be carried through explicitly. The results of Meixner [7] are used as a 
guiding principle for the successive approximations. The method is discussed 
in section 5. The mathematical aids are given in the appendix. 

Notations 

The system of units is the '^practical system"; i.e., the electric field strength 
is measured in volt/cm etc. The frequency co of the incoming waves defines 

Paper presented at the June, 1950, Symposium on the Theory of Electromagnetic Waves, under 
the sponsorship of the Washington Square College of Arts and Sciences and the Institute for 
Mathematics and Mechanics of New York University and the Geophysical Research Directories 
of the Air Force Cambridge Research Laboratories. 

393 (S39) 



394 (S40) ELECTROMAGNETIC WAVES 

the wave length X = 27rc/w, (c = velocity of light) ; a, h are the measures of the 
wave guides, and we define: 

k = ko = 2t/\ 

k^ = ^'o[l - m'\'b-Y\ m = 0,1,2, '" 

km is negative imaginary if m ^ 1. 

Ai = \/h in Part one; (/u > 1) 

km^^y2 = kll - (m + iyx'a-T\ m = 0, 1, 2, -- - 

ki/2 is real and positive, km +1/2 is negative imaginary if 

m ^ 1 

M = \/a in Part two; (2 > )u > 2/3) 

j8 denotes the relative width of the diffracting strip (as compared ^ith 6 or a in 
§2 or §3). 

8nm = on n 9^ m, 5„,„ =1; n, m = 0, 1, 2, • • • . 

I = (5„,m) is the unit-matrix. 

€„ = 2 if n = 1, 2, 3, . . . ; €0 = 1; ia)n = T{a + n)/T{a) 



I. A Diffracting Strip between Two Parallel Planes 

1. Elementary Results 

We consider two parallel planes which, in Cartesian coordinates x, y, z are 
defined by the equations z — d=|6. Between the two planes we have an infinite 
strip defined by re = 0, — i/36 ^ z ^ \0b, — co<:y<:co;0<(3<l. A cross 
section of this arrangement is shown in Figure 1. 






■^X 



Figure 1 

We assume that the planes and the strip have infinite electric conductivity 
and we consider an incoming electromagnetic wave which is defined by 

(1.1) - E^E,e-''% 



W. MAGNUS AND F. OBERHETTINGER (S41) 395 

where E denotes the ^-component of the electric field, Eq is a constant, and 
where the other components of the electric field vanish. The time factor exp 
[icct] will be omitted everywhere. 

We may assume that the x and y components of the diffracted wave vanish 
and that its ^-component E^ can be expanded in a series 

00 

(1.2) EXx, z) = -GOtt J2 ^nikjk) cos {2Trmz/h) exp {-i\x\ K] 
which is convergent for all x y^ 0. If the series 

00 

(1.3) I{z) = Y^Am cos 2Trmz/h 

nl = 

converges, we may call I(z) the current on the strip. We could derive the ex- 
pression for E^ from I{z) and from the formula for the field of an electric dipole 
between two parallel planes (cf. for instance [8]). 

From the condition that the total energy of the field contained in any 
finite part of the space should be finite we find that 

(1.4) If \E.\'dxdz=\Ao\' + ^f:\ Al I J' exp {-2 | xK II I K \' dx 
is bounded if e ^ 0. Therefore we liave 

(1-5) EMU |fc„| <co. 

m=l 

The boundary conditions are 

(i) EXO, z) = -Eo for -^jSh < z < |/36 

(ii) I(z) =0 for i/36 < I H < i^; 

Condition (ii) can be derived from the fact that I{z) is (apart from a constant 
factor) the ^/-component of the magnetic field at a; = 0. It must be regular 
outside the strip. From Maxwell's equations it follows that I(z) is an odd 
function of x since the electric field is an even function of a;. If x ^ oo we assume: 
(iii) There exists a constant C such that 

(1.6) lim I EXx, z) - C exp l-ik\x\} \ = 0. 

Ul-oo 

In the present very simple case this happens to be equivalent to Sommerfeld's 
''radiation condition [4, 10, 13], because we assume that only ko = k is real and 
that km is negative-imaginary f or m = 1, 2, 3, • • • . 
From (i) and (ii) we find 

(1.7) f' Iiz)lEXO,z)+Eo]dz = 0. 
J-hh 



396 (S42) ELECTROMAGNETIC WAVES 

Since E,{x, z) must approach £^^(0, z) if a; — > ^ we deduce from ('1.2) and 
(1.5) that we can introduce the formal series for £^^(0, z) and /(z) into (1.7). 
This gives 

(1.8) eOTTilo^o = (Re^o^o 

00 

(1.9) SOtt E I ^^ I ^^X = ^m A;Ao^o • 

m = \ 

Therefore the A^ are uniquely determined by (1.5) and by the boundar}^ condi- 
tions. If this were not true, a non-trivial solution of the problem would exist 
for which E^ = 0; this contradicts (1.8), (1.9). Finally, the complete solution 
of the problem is uniquely determined by (iii) if the A^ are given. 

If \x\ — »co. only the first term in the expansion (1.2) contributes to E^ . 
Therefore we may call 

(1.10) R = -mwAo/Eo 

the ''reflection coeflScient". It can be sho^\Ta (in an elementary way) that | R \ 
is always different from and 1 if /3 (the relative ^^idth of the diffracting strip) 
is different from and 1. But the inequalities for | R \ which can be obtained 
by an elementary method are unsatisfactory. 

2. The Linear Equations for ^ = ^2 

From (1.5) it does not follow that the expansion (1.2) is convergent if 
X = 0. In order to obtain a system of linear equations for the A^ we shaU 
assume that this is the case. This assumption can be justified to some extent 
by the results in [7]. According to ]\Ieixner, 

limEX0,z)[i(3'b' -zT" 

exists a z = zki(3h d= e and e ^ 0. Since we can construct a convergent Fourier 
series for a function which is constant in one of the domains J/36 < \ z \ < J6 
and — i/36 < z < ^(3h and is equal to 



lifb' - ^1 



■1/2 



in the other domain, we may expect that E^{0, z) can be expanded in a series 
of this type plus the Fourier series of an L^-function which is at least of the class 
C except at z = zhi^b. This and the formulas (A. 18), (A. 19) of the appendix 
lead to the assumption that 

(2.1) I A^m''' I ^ M 

where M is independent of m. 



W. MAGNUS AND F. OBERHETTINGER (S43) 397 

Normalization. We denote z/h by f and we take Eq = GOtt. Then — Ao is the 
reflection coefficient. We define a complete orthonormal system of even func- 
tions 0^(rt in (-J, i) by 

(2.2) 0o(r) = 1, 4>n.{^) = V2 cos 27rmr, m = 1, 2, 3, • • • 

According to H. L. Schmid's lemma (cf. Appendix I) we find that the A„ 
have to be computed from 

(2.3) ^n = S Un.mXm; Un,ra = / cf>,,{^)cl>m{^) ^^ 

where the x^ are determined by the equations 

T CO 00 

(2.4) ^ J2 Un,mXm + Z (^n.. " U^,m)Xm = h ,n , 71 = 0, 1 , 2, ' ' ' . 

Restriction to ^ = \, and Approximation. If /? = J, we may write Un,m — \^n,7n + 
h^n,m , where the matrix S = (Sn,^) satisfies aS^ = I. Introducing 

(2.5) yi= V'^ii-lfx^i , ir = {-iyx,r^, , l,r = 0,1,2,'" 

and computing the 5„.^ from (2.3) we find that (2.4) can be written in the form 

(2.6) 7o = 1 

^''^ -^^ ^^ ;^2 S (Trfri + ^Ti)- = ^' ^ = 1,2,3,... 
^'-'^ '-'^-'^^kUrfhr^.^ -0,1,... 



where 



1 I ■^'^ r-i -2 -2il/2 ^ , { A\ 

T« = T =1+ [1 — m fl ] 0-2 IM = TJ, 



km- k 



(2.9) 

m = 1, 2, 3, ••• . 

We can express the yi in terms of the T]r by applying Titchmarsh's inversion 
formula (cf. Appendix I, (A.7), (A.8)) to (2.8). This gives 

Z = 0, 1, 2, • • • ; €o = 1; €i = €2 = " ' = 2, 
Eliminating the ji , we find from (2.6), (2.9), (2.10): 



398 (S44) ELECTROMAGNETIC WAVES 

(2.11) ^±'-^=-1 

(2-12) Z {^TTT+l + 1 + 1+ h i^'" - ^^-)''' =^^' ' = i> 2, 3, • • • 

Expanding the r^ in a series of powers of (mfj.)'^ and neglecting the terms 
of a degree greater than two, we obtain the Approximative set of linear equations: 



y ^oT + i 



1 + 



i 111 1 



(2.13) 



W !^ /i_i_ri__i_ 

Vtt :eo -Z + r + iV 2m L^ r + i 



= 0, Z = 1, 2, 3, 



We apply the Linfoot-Shepherd inversion formula for d = J(cf. Appendix I) by 
multiplying the Z-th equation by 

r + 2/ n! -l + n + i U ' » - "' 1' A 

and adding all the equations. This and an application of the fonnulas in 
Appendix IV gives 



(2.14) 



"» + ^ n! M l^ "i r + J + (r + §)^ + 2m (r + if,} 

I 1 1 v_jlI i^(i)» 



V2 
We can obtain a solution of (2.14) by letting 

Defining S and T by 

(2.17) r=i:r^, 

they become linear functions of <r, t if we substitute the right side of (2.13) in 
(2.16), (2.17); then equation (2.14) can be written in the form 



(2.18) 



1 1 iV2( 1 1 ) 

+ 2;m"~M~v^2« + iV-^- 



W. MAGNUS AND F. OBERHETTINGER (S45) 399 

This is satisfied (for all values of n) if and only if 

(2.19) ,-i^S=l, r-'-^T = 0. 

H T fJL T 

Therefore we obtain two linear equations for o-, r, which can be written in the 
form 



<2.20) 






(2.21) ,i.^^ + ,(l+_i_^^).0 
where (cf. Appendix IV, A.23) 

(2.22) 5„= |:®^(r + |)"", n= 1,2,3, .•• 

(2.23) S, = tt; S, = 27r log 2; ^3 = 27r[(log 2)' - 3^ tt^J. 

Equations (2.20) and (2.21) determine a, t uniquely; therefore the rjr are given 
by (2.15). From these approximate values of the r/^ we obtain approximate 
values of the yi from (2.6), (2.7). From (2.3) and from the equations (2.7), 
(2.8) we can now compute the y4.„ , n = 0, 1, 2, • • • . This gives 

Theorem 1. If ice simplify the linear equations (2.6), (2.7), (2.8) hy sub- 
stituting for r^ in (2.9) the first three terms of its expansion in a series of powers 
of (/im)~\ then the Fourier-coefficients of the ''current on the diffracting strip 
become 

(2.24) Ao = i - i(^ + T log 2) 



(2.25) 



(2.26) 



1 = 1,2,3, ■■• 

r = 0, 1, 2, • • ■ 



400 (S46) 



ELECTROMAGNETIC WAVES 



where a, r are defined by (2.20), (2.21). For large values of fi we have approxi- 
mately 



1 



1 + fx-H log 2 



+ 0(m-'); r = -^ (l + ^ log 2)" + 0(m-'). 



The approximate solutions A^ in equations (2.24), (2.25), (2.26) satisfy 
(2.1); the reflection coefficient Aq —> ii ^ -^co. 



II. A Diffracting Strip in a Wave Guide with a 
Rectangular Cross Section 



3. Elementary Results 

Let us consider a rectangular wave guide which extends in the direction of 
the a;-axis. The corners of the cross section of the wave guide in the ?/,2;-plane 
are given by y = =bja, 2; = ija', a diffracting strip occupies the area — J/Sa ^ 
y ^ i(3a, — ia^ ^ z ^ ^a' , where < /? < 1; in §4, we shall choose /? = |. 
Figure 2 shows the cross section of the wave guide. 







\ 


z--/^' 


if_ 


:i^°. 


w. 





Figure 2 



We assume that the wave guide and the (infinitely thin) strip have infinite 
electrical conductivity and we consider an incoming wave which is defined by 



(3.1) 



E = Eq cos (iry/a) exp {—ixki/z}, 



where E denotes the ^-component of the electric field, Eq is a constant, and all 
the other components of the incoming electric field vanish. The time factor 
exp {io:t} will be omitted everywhere. 

As in §1, we can show that the diffracted wave has an electric field parallel 
to the 2-axis, and that its ^-component can be expanded in a series 

00 4 
(3.2) EXx, y) = -m-irk X t^^ cos (2m + 1) ^ exp {-i\x\ K^u2} 

m = i^m + l/2 ^ 

which converges for all | a; | > (and even for a; = 0, as we shall see). If the 
series 



W. MAGNUS AND F. OBERHETTINGER (S47) 401 

00 

(3.3) I{y) = E ^". cos ((2m + l)iry/a) 

m = 

converges, we can call I{y) the current on the strip and derive the diffracted 
wave from (3.3) and the formulas for the electric field of a dipole in a wave guide, 
(cf. [8]). Apart from a constant factor, I(y) is the ^/-component Hy of the mag- 
netic field of the diffracted wave; we have 

(3.4) I{y) = if i^a<\y\ <ia. 
The other boundary conditions are 

(3.5) ^.(0, y) = -Eo cos ^ for -| ^a < ?/ < | (3a 



(3.6) lim 



|x|-»oo 



Ez{x, y) — C cos -^ exp { — ^ | a: | k^, 



for a suitably chosen constant C. This is not the radiation condition of Som- 
merfeld which, in general, cannot be satisfied in a wave guide, not even in the 
form allowing the boundaries to extend to the infinite parts of the space, (see 
Rellich [9]). 

The condition of the finiteness of the total energy in a finite part of the 
space gives 

(3.7) i: M. n^^.i/2 V <-. 

This can be shown by integrating the square of the ^/-component of the mag- 
netic field over the volume. As a consequence of (3.7) we see that the series 
in (3.2) can be used for the representation of E^ \i x = 0. As in §1, we can 
show that 



(O.o) ^p. , Aq — AqAq + lki/2 2^ AmAm/ \ km + 



1/2 



This proves the uniqueness theorem. If the. A^ satisfy (3.7) they are uniquely 
determined by (3.4), (3.5). The complete solution is also uniquely determined 
because of (3.6). 

The reflection coefficient becomes 

(3.9) R = -f^ A, . 

Jl/oKi/2 

It can be shown that | i2 | is always different from and 1 ii (3 9^ 0, (3 ^ 1. 

We can also prove an inequality which | R \ must satisfy. 

Since 

(3.10) r I EXO, V) r dy § ^ \ E, \' (cos "fj dy 



402 (S48) ELECTROMAGNETIC WAVES 

it can be shown (from (3.8), (3.2)) that 

(3.11) I 72 1^ ^ ^ + 1 sin ^^ - I (l - 1 ^^''\ 
where 

(3.12) n = X/a; 2 > ^ > 2/3. 

4. The Linear Equations for P = ^ 

We normalize Eq in such a way that 

^^•^^ E~k~ = ^ 

and we introduce f = iry/a as a new variable. The functions 

(4.2) 0^(r) = V2 COS ((2m + l)7rr) 

form a complete orthonormal set of even functions in — J ^ ^ ^ |. The 
equations to be satisfied are (if /? = i) : 

(4.3) i: A^Ui) = if +j < I r l< J 

(4.4) i: 4„ T^i^ *„(f) = 0„(r) if - J < r < J. 

According to Meixner [7] we shall have to expect that the series in (4.3) becomes 
infinite like ( — f^ + 1/16)"^^^ in ( — J, J); from (A.19) in the Appendix we see 
that in this case 

(4.5) A,: = CJ-''' + ein, 1=1,2,3,'" 

(4.6) I A,r.,r I ^ C2 , r = 0, 1, 2, . . . 

where Ci , C2 are constants. 

From H. L. Schmid's lemma (Appendix I) and from (4.3), (4.4) we find 
the linear equations 

(4.7) Xn+ Sn J^ Sr,,mXm = do ,n 

where 

f^l/2 T^ '^n + 1/2 






M(n + 1/2) \^ " y ' 2/ y M^'(?t + 1/2)' 

1 - (2ra + 1)-' 



W. MAGNUS AND F. OBERHETTINGER (S49) 403 

/1/4 
-1/4 

and where the original unknown quantities A^ are given by 

(4.10) ^n = I Z (5n.n. + S^,Jx^ . 

Introducing the new variables 

{-iyx2i = 7/ , (-l)''a:2r+i = Vr , Z, r = 0, 1, 2, ••• 
and evaluating the s„,^ from (4.9) we find 

(4.11) To = 1 

(4.12) y, + d.^r-' ± TTVT^ + ^-^" ^ 7 /^; ■ 1 =0, 2 > 

r = t- r ' T" 2 r = t* "T" ' "T 2 

(4.13) ,, + e..,„r-' i; , 7' , , - fl.,..x-' i; , , !' 3 = 0, r S 0. 

We can use (4.13) in order to express the ji in terms of the rjr - For this purpose 
we have to apply the Linfoot-Shepherd inversion (cf. Appendix I) in the case 
where the value of the parameter 6 in (A. 10) is —J. The inversion formula is 
not unique in this case, but the yi are uniquely determined if we apply (A. 10) 
in a formal way, as will be seen later. That this is permitted could also be shown 
by a closer anatysis of (4.12), (4.13). We can eliminate the yi by using (4.11), 
(4.12). The result is the following set of hnear equations for the r]r in which 
we did not yet neglect any terms whatsoever: 

(4.14) i:t{-^.-;v. + :-$fV = i 

We expand the dm in (4.8) in a series of powers of m"\ m = 1, 2, 3, • • • , 
neglecting all terms of order m~^ and of higher order. Substituting these ap- 
proximations for the 021 , dzr+i in (4.14), (4.15) we obtain the approximate 
equations: 



(4.17) 



Z^fr + i 



r + 7 I „ _L 1 - nnh' = 0' 



2/l-i + r + J • l + r+ 1 r + i 



1, 2, 3, 



404 (S50) 
where 

(4.18) 



ELECTROMAGNETIC WAVES 



if(i-VM 



2 > M > 



We now apply the inversion formula (A. 14) in the appendix to (4.16), 
(4.17); the result is 



(4.19) 



„ , (J),y> (i)s{s + i) [ a _,_ i 1 I (^). 

'^^■^ r! h^' si L(s + J)(s + i)"^s + fJ 2(r+l)r 



r = 0,l, 2, 



In deriving (4.19) we have used the formulas (A. 20) to (A. 22). B}^ a repeated 
application of these formulas and by putting 



(4.20) 



Vr = C 



(i)r 



(r+ 1)! 



we find for the constant C the value 

(4.21) c = l{i + 16.A + (I - 2.)[r(|)/r(f)]|'. 

In order to determine the ji , we us again (4.13) to express the ji in terms 
of T]r . Combining this with (4.12), we find 



(4.22) 



^ =-■(-!)['-; 4) 



where the matrices A ( — J), /, H(§) are defined as in the Appendix (A. 6), and 
g, h, denote the vectors with the components 



(4.23) 



(1 + e^!)y, 



(1 + d;r\i)vr 



If we substitute again 

(4.24) -1 - 4(7/(m + i), 



iv,.-i 



iM-(l - m74)' 



for eZ\ TO = 1, 2, 3, • • • , we obtain from (4.22), (4.20) and from (A.22) 



(4.25) 



r(i)V (i). 



- = ^W1^' '='''''' 



Now we can compute the coefficients A„ from (4.10), (4.20), (4.21), (4.25), 
(4.11), (4.12), (4.13). The result is 

Theorem 2. If we substitute -1 ± 4:am~^ for C' in (4.12), (4.13), (4.22), 
we obtain the following approximate values for the constants A„ which determine 
the diffracted wave: 



W. MAGNUS AND F. OBERHETTINGEE (S51) 405 

(4.26) ^» = 2 + ; + ^;r 1+07+^ 

(4.27) ^. = (-l)'(l - e-) Y^T^V^ f > ^ = 1, 2, • • ■ 

(4.28) ^... = (-l)'(l - ^.-.,) TITq^ (TTl)! ' r = 0, 1, . . . 

(4.29) Q. = {I r(|)/r(|)y, Q. = '-l- m. 

(4.30) ^ = i^V'(l - iu74)''', iu = X/a, f< m < 2 
and i(;/ie7^6 ^/^e ^^ , m = 1, 2, 3, • • • , are gf^^;e?^ 5?/ (4.8). 

The An satisfy (3.7). 

Although Aq (which now is the reflection coefficient R) satisfies the in- 
equahty (3.11), the approximation (4.26) is unsatisfactory since ^o -^ 1-07 
(instead oiAq— >l)ifo-— >0. This is due to the fact that the expansion of 6^ 
in a series of powers of m~^ is not also an expansion in a series of powers of cr. 
Therefore the higher terms of this expansion would contribute to the terms of 
Aq which do not involve o-. 

5. Concluding Remarks 

(i) Higher Approximations, The formulas of theorems 1 and 2 may be 
characterized as a second and a first approximation respectively, according to 
the number of terms in the expansion of the r^ , d^ which have been kept for 
the final setup of the linear equations. If we wish to deal with higher approxi- 
mations, it is not necessary to develop any new methods. It can be shown that 
the solution of the original (exact) system of linear equations involves the in- 
version of a bounded linear operator T at a point of its spectrum which lies on 
the boundary of the spectrum of T^T (where T* denotes the adjoined operator). 
The Linfoot-Shepherd inversion for ^ = ±i is an inversion of this type. After 
its application the solution of an approximation of finite order of the given 
system of linear equations requires only the inversion of linear operators which 
have a bounded inverse and the solution of a finite system of linear equations. In 
the case dealt with in §2 it can be shown that (at least for a sufficiently large ^u) 
the solutions for the n-th approximation tend towards the solution of the exact 
system of linear equations. By a different method, Lamb [2] has obtained a 
first approximation for the solution of the problem of §2. However, it seems 
that his method does not lead to an approximation of a higher degree. 



406 (S52) ELECTROMAGNETIC WAVES 

(ii) The Restriction i3 = J. It can be shown that the Linfoot-Shepherd 
inversion with ^ = J is equivalent to the solution of the first boundary value 
problem of Laplace's equation in two dimensions, where the boundary consists 
of all the intervals 

(5.1) x = 0, n-i^y^n-i-i, n = 0, d= 1, ± 2, • • • 

of a straight line (the y-axis) and where the boundary conditions are periodic 
and of period one. This explains why the Linfoot-Shepherd formula can be 
applied to the problem of §2, which, f or ju — > oo , reduces to a problem connected 
vnih Laplace's equation. It is to be expected that a solution of the corre- 
sponding boundary value problem ior (3 y^ ^ will lead to the right generalization 
of this inversion formula. 

(iii) The Case of a Diffracting Thin Wire. We can replace the diffracting 
strip by a thin wire of radius p, the length of which equals the width of the strip. 
In this case the coefficients of the linear equations (2.4) or (4.12), (4.13) must 
be changed in the following way: 

In (2.4) substitute 

(5.2) iklJo{Kp)H'o'\Kp), m = 0,1,2, '" ; h = k, 

for km , where Jo , Hq^^ denote Bessel functions of the first and third kind re- 
spectively. 

Instead of (4.12), (4.13) use again (2.4) and replace km by 

(5.3) iklJo{kmP)Smip, a, d) 

where d denotes the ^/-coordinate of the axis of the ^\'ire and where 

(5.4) S^ = Hl'\kmp) + 2 i: m'\2ank^) - E Hi'\\ (2n + l)a + 2d \ /bj. 

n=l n = — oo 

The quantities which are to be substituted for the k^ have the same asymptotic 
behavior as the /b^ f or m — > oo , and therefore a first approximation for the linear 
equations can be derived and dealt with in the same way as in the case of a 
diffracting strip. For the uniqueness theorem cf. Schaefke [10]. The S^ have 
been tabulated, because they play a role for the computation of the effect of a 
diffracting wire which connects the opposite boundaries of a wave guide. 



Appendix 

1. H. L. Schmidts Lemma 

Let <l)m(x), m = 0, 1, 2, • • • , be a complete set of orthonormal functions for 
the interval (0, 1). Let /J be a real number, < (3 < 1 and let 



W. MAGNUS AND F. OBERHETTINGER (S53) 407 

(A.l; E A^cf>^{x) = if <x <1 

(A.2) E K^A^cl>^(x) = Ux) if <x <^. 

We define 

(A.3) Un,m = / <f>n{x)4>Jx) ClX] Vn,m = / (i>n{x)<j)Jx) dx. 

Then the matrices U, V with the general element Un,m , i^n.m satisfy 

(A.4) UV = VU = 0; [7+7 = 7; U' = U, V = V, 

where / denotes the identity. If we denote by a the vector with the com- 
ponents An , then there exists a vector x such that 

(A.5) a = Ux, {V + KU)x = e 

where e = (1, 0, 0, • • •) and where K denotes the diagonal matrix (5n,^K:^). 
This statement is vahd if the functions on the left hand side of (A.l), (A.2) are 
absolutely integrable and if it is permitted to multiply (A.l), (A.2) by 0„(x) 
and to integrate the left hand side term by term in (0, jS) and {(3, 1). A proof 
can be derived from Schaefke [10]. 

2. Inversion Formulas 

Let ^ be a real parameter, ^ ?^ 0, — 1, — 2, • • • and let A{d), H{d) be the 
matrices ^vith the general elements 

, . sunrd 1 



TT —n -\- m -{- d^ 
(A.6) 

h^-M =n + L+0' ».»» = 0,1,2, ••• 

where n denotes the row and m denotes the column of the matrix elements 
dn.m , K,m • The matrices A{d), H(6) are bounded; cf. [11]. The following is a 
special case of Titchmarsh' s inversion formula [13]: The matrix 

(A.7) ^ = 2-^(1) + ;^^© 

has a bounded inverse T~^ the general element of which is 

(A.8) 7= \ r-^ ; — r H ; — - — ; — rf; eo = 1, €i = €2 = — =2* 



408 (S54) ELECTROMAGNETIC WAVES 

Titchmarsh has shown, that under very mde conditions for a vector y 

(A.9) T-'y = X if Tx = y 

and vice versa. 

The Linfoot- Shepherd inversion formula gives a formal inverse A~^{d) of 
A{e). The general element of A"\^) is 

_sin^ r(l+n+.) 1 ni + m-e) ^ ^ .... 

TT n! — n + w — ^ m! 

Linfoot and Shepherd [3] stated certain sufficient conditions for a vector y such 
that 

(A. 11) A-\e)y = X if A{d)x = y. 

They also showed that for ^ ^ the homogeneous equations 

(A.12) A{e)x = 

do not have any solution whatsoever except x = and that for — 1 < < 0, 
(A.12) has the only non-trivial solution x = {x^} where 

ml 

Here C is an arbitrary constant and 

. -, T(u + m) 

(A.13) ^ ^ 

= u{u -\- 1) ' - • (u + m — V) if m = 1, 2, • • • ; (u)o = 1. 

It can be shown that A~\e) is bounded if and only if — J < ^ < i cf. [5], [10]. 
A result connected ^vith (A. 10) is: 

The inverse of 7r^(J) + H{1) = G is a hounded matrix G~^ = (^„.,„) the 
general element of which is 

The boundedness of G~^ follows from (A. 14) because 

- I + I 

—n +m + i n+m+l 
<A.15) 

_ _2m 

2n + | 
and 



i/ 3I + I I 

i {—n +m — J n+m+ IJ 



(i)n 



(A. 16) lim ^ V^ = 7^-^'^ 

„_oo n\ 

This shows that the elements of G~^ can be obtained from those of ^(1) — 



W. MAGNUS AND F. OBERHETTINGER (S55) 409 

7rA( — J) by multiplying them by certain bounded positive factors; the rest 
follows from a criterion of I. Schur [11]. The proof of (A. 14) can be obtained 
from [5]. It is not difficult to prove that G^G has a minimum >0 and that 
therefore G has a bounded inverse. But although (A. 14) is a formal inverse 
of G it is still necessary to prove that it is bounded; cf. [5]. 

3. Some Special Fourier Series 

From the Hansen-Bessel integral representation for the Bessel function Jq 
of the first kind 

(A.17) Jo(z) = X-' f^ (f^p dt 

we find that 

(O if J/3 < a; < i 

(A. 18) 2 €mTrJo(Trm^) cos 2Tmx = \ 

((i/3^ - xy' if - 



The convergence of the series on the left hand side in (A. 18) can be proved by 
expressing the partial sums as an integral which can be derived from (A.17). 
The behavior of the coefficients in (A. 18) if m — >co is given by 

Joimir^) = '^~\-^''' cos (mTT^ - i)[l + OimT')]. 

The formula 

((2 cos27rx)-'^' if -i < a; < i 



(A.19) E (-1)'^ % cos {7r(4n + 1)^1 = 



.0 if i < U I < i 



can be proved by expressing the series in terms of two hypergeometric functions 
of argument exp { d= i^irx ] . 

4. Sums 

The summations which are involved in the multiplication of infinite matrices 
in sections 2 and 4 can be carried out explicitly by using the following formulas 
and their derivatives with respect to z: 

(A.20) :J?^=i:^^-=T^"^, (cKe.>0) 



(A.21) 



rfflv r(^) _ r(r) 



^S'^fe-Fi.}. (->-'> 



410 (S56) ELECTROMAGNETIC WAVES 



(A.22) 



r(l - 26 + 2a)r(l - 26; T(a - z)T(a + z) 

T(2a) r(l + a - 26 - z)T(l +_a -26+2) 

_ f. (2a).(26). J 1 , 1 y ,^ , . 1, 

~ ;^ (1 - 26 + 2d),n\ [a -\- n - z'^ a + n + zj' ^^^ ^ 2; 



Equations (A.20), (A.21) follow from the expansion of Euler's Beta-integral in 
an infinite series. For a proof of (A.22) cf. [5]. The sums *S„ in (2.22) have 
the generating function 

and the first three of them can be expressed in terms of ir and log 2 by using 
well known properties of the Gamma Function. 



Bibliography 

1. C. J. Bouwkamp, On the freely vibrating circular disc and diffraction by circular discs and 

apertures, Physica, Volume 16, 1950, pp. 1-16. 

2. H. Lamb, On the reflection and transmission of electric waves by a metallic grating, Proceedings 

of the London Mathematical Society, Volume 29, 1898, pp. 523-544. 

3. E. H. Linfoot and W. M. Shepherd, On a set of linear equations (II), Quarterly Journal of 

Mathematics, (Oxford Series), Volume 10, 1939, pp. 84-98. 

4. W. Magnus, Uber Eindeutigkeitsfragen bei einer Randwertaufgabe von Am -h khi = 0, 

Jahresbericht der Deutschen mathematiker Vereinigung, Volume 52, 1943, 
pp. 177-188. 

5. W. Magnus, Uber einige beschrankte Matrizen, to appear in Archiv der Mathematik. 

6. A. W. Maue, Zur Formulierung eines allgemeinen Beugungsproblems durch eine Integral- 

gleichung, Zeitschrift fiir Physik, Volume 126, 1949, pp. 601-618. 

7. J. Meixner, Die Kantenbedingung in der Theorie der Beugung elektromagnetischer Wellen an 

vollkommen leitenden Schirmen, Annalen der Physik (Series 6), Volume 6, 1949, 
pp. 1-7. 

8. F. Oberhettinger and W. Magnus, Uber einige Randwertprobleme der Schwingungsgleichung, 

Journal fiir die reine und angewandte Mathematik, Volume 186, 1945, pp. 1-9. 

9. F. Rellich, Uber das asymptotische Verhalten von Losungen von Au -{- \u = in unendlichen 

Gebieten, Jahresbericht der Deutschen mathematiker Vereinigung, Volume 53, 

1943, pp. 157-165. 
10. F. W. Schaefke, Uber einige unendliche lineare Gleichungssysteme, Mathematische Nachrich- 

ten. Volume 3, 1949, pp. 40-58. 
IJ. L Schur, Bemerkungen zur Theorie der beschrdnkten Bilinearformen, Journal fiir die reine 

und angewandte Mathematik, Volume MO, 1911, pp. 1-28. 

12. A. Sonmierfeld, Die Greensche Funktion der Schwingungsgleichung, Jahresbericht der 

Deutschen mathematiker Vereinigung, Volume 21, 1912, pp. 309-353. 

13. E. C. Titchmarsh, A series inversion formula, Proceedings of the London Mathematical 

Society (Series 2), Volume 26, 1927, pp. 1-1 L 



Wiener-Hopf Techniques and Mixed Boundary 

Value Problems* 



ByS.N.KARP 
New York University 

Introduction 

J. Schwinger [1] has shown how Green's theorem may be utihzed to formu- 
late certain of the boundary value problems of electromagnetic theory and 
acoustics relating to the equation Au + k^u = 0, as Wiener-Hopf integral 
equations, the boundaries in question being, in general, semi-infinite planes or 
cylinders. The Wiener-Hopf equation /(x) = /" g(xo)K{x — Xq) dxo : x > 0, 
is solved by the application of the Fourier transform, together mth function- 
theoretic considerations in the Argand plane of the transform variable. Fourier 
transformation results in a single equation between two unknown transforms, 
and this equation is solved for both unknown functions by function-theoretic 
techniques. The notable success of this procedure in the hands of Schwinger [1], 
Carlson and Heins [2], Heins [3], Levine [4], and others has stimulated further 
investigation. G. Carrier [5] has indeed sho^vn that the procedure may be applied 
to other partial differential equations. He has illustrated this point by con- 
sidering a boundary value problem for a generalized Tricomi equation, 

Uyy + y^iu^x + k^u) = 0, 

and a semi-infinite plane obstacle. The excitation was taken to be a ^'plane 
wave," i.e., a function reducing to the latter when m = 0. The integral equa- 
tion resulting from Green's theorem is not of Wiener-Hopf type, but when a 
certain Hankel transformation was applied (reducing to the Fourier transform 
for m = 0), a relation between two unknown transforms resulted, which was 

Paper presented at the June, 1950, Symposium on the Theory of Electromagnetic Waves, under 
the sponsorship of the Washington Square College of Arts and Sciences and the Institute for 
Mathematics and Mechanics of New York University and the Geophysical Research Directories 
of the Air Force Cambridge Research Laboratories. 

*This work was performed at Washington Square College of Arts and Science, New York 
University, and was supported in part by Contract No. AF-19(122)-42, with the U.S. Air Force 
through sponsorship of Geophysical Research Directorate, Air Force Cambridge Research 
Laboratories, Air Materiel Command. 

411 (S57) 



412 (S58) 



ELECTROMAGNETIC WAVES 



sohxd (after an ingenious change of complex variable) by the customary tech- 
niques/ 

In the present note the parallelism between the method of separation of 
variables and the Green's function integral equation method is shown to persist 
in the present situation as it does in problems of more classical type. This re- 
lationship leads to a characterization (from the standpoint of coordinate systems) 
of those problems in which the ^'Wiener-Hopf" type of problems (in an extended 
sense) arise. Certain heuristic advantages of the separation of variables pro- 
cedure are also pointed out. 

(1) To fix the ideas a simple and typical illustration of the integral equation 
method of Sch^vinger is recalled at this point. The problem relates to the diffrac- 



\^2 



— £-»> J 

ii > t I ) ) , > , 



aScreen 



—>X 



•v^ 



ji 



Figure 1 



tion of a plane wave by a semi-infinite plane; it has also been solved in the 
present manner by Copson [6] independently of Sch^^dnger. The mathematical 
problem is to find a function ii{x, y) such that, 

(1.1) u^^ -\- Uyy -{- k^u = 0, where 



(1.2) 



dy/y=o 



k = k, + ik2 , 1 » A'2 > 
for X > 0, 



(1.3) 
(1.4) 
(1.5) 



U = Uo -^ III , 

Uq = exp {ik{x cos do -\- y ^m 6^}, 

as r -^ CO , off the positive x-axis, 



(1.6) 



u regular in exterior of the positive 3;-axis. 



Here x and y are Cartesian coordinates, r and 6 the usual polar coordinates, 
and do gives the direction of the incident field Uq . 

^The present investigation was suggested by this work and by a private remark of this 
author that the transform to employ is suggested by the form of the product solution of the 
partial differential equation, in the ioTin.f{x)g(y). 



S. N. KARP (S59) 413 

The free space Green's function is introduced. It is 

G(x, y: xo , yo) = \ Hl,'\k{{x - x,f + (?/ - y,)^] 

where Hl^^ is the Hankel function of the first kind. Green's theorem is apphed 
to the functions u and G, along the contour c = Ci + Cg indicated in Figure 1; 
Ca is an arbitrarily large circle. 

Using the singularity of G at {x, y) = (xq , yo) , and the boundary conditions 
(1.2) and (1.6), and letting Cg — »oo, one finds 

u(x, y) = [u]-T- (x, y:xo ,yo) dxo 

'Jo oyo J2/o=o 

(1.7) 

+ exp {ik(x cos do -\- y sin do)] 

where [u] is the discontinuity in u across the screen. Differentiating mth respect 
to y and applying (1.2) again one finds 

(1.8) = ik sin ^o exp {ik x cos ^o) + / [u{xo)]K(x — Xo) dxo , x > 0, 

where 

d'G 



K{x — Xo) is 



dy dyo 

This is a typical inhomogeneous Wiener-Hopf equation. One now defines 
g(x) = 0, X > f{xo) = [u], Xo > 

(1.9) h{x) = 0, x <0 fixo) = , a:o < 

h(x) = ik sin do exp {ik{x cos ^o)!, a; > 
and then equation (1.8) may be rewritten, 

(1.10) g{x) = h{x) + f f{xo)K{x - Xo) dxo , 



— oo < X < 00 



in terms of the two unknown functions g{x) and f{xo). Multiplying by 
exp { —tax] and integrating \A'ith respect to a, one finds 

(1.11) gia) =Ka)+'j(a)K(a) 

where for instance K{a) = jZoo K(t) exp { —iat} dt. This equation holds in the 
common strip of regularity (provided such exists) of the functions g{a), /(a), 
K{a), h{a) considered as functions of a. 

The technique of determining the regions of regularity of these functions is 



414 (S60) ELECTROMAGNETIC WAVES 

as follows. One assumes for/(a:o) say, a behavior like exp {ikXo} for large Xq , 
where ^m{K) > 0. Then from the formula 

(1.12) J((x) = / f(xo) exp {-iaxo] dxo 

we find that J (a) will be regular in the lower half -plane ^m{oi) < ^m(K), which 
starts above the real axis in the a-plane. One finds that h{a) has similar be- 
havior, while g(a) is regular in an upper half -plane starting below the real axis. 
Also 

K{a) = I (/b' - ay'\ regular for | ^m(a) \ < ^m{k). 

Hence in a strip enclosing the real axis, we have, 



2\l/2 
J 



the first term on the right being h-(a). The subscripts +, — , denote regularity 
in upper and lower half-planes respectively. 

K(a) is now factored in the form K-{a)/K+{a) where K-{a) = (k — a)^^^^ 
and K+{a) = {k + a)'^^^. After multiplying by K+{a) and applying suitable 
manipulations, one finds 



g+{a) k sin ^p \ 1 



{k + ay a - k cos do l{k + aY'' {k -\- k cos 
(1.14) 



cos 60^'] 



k sm ^0 I " ^ / \/7, \i 

+ o f-{c^){k - a) 



(k -\- k cos doY {a — k cos ^o) 2 

an equation whose left and right sides are regular in upper and lower half-planes 
respectively. Since these half-planes overlap, an entire function, E{a) say, is 
defined by (1.14), the left and right sides being various representations. The 
growth of this function at infinity is studied via the growths of g+ia), and 
f-{a), and it is found to be of negative fractional degree; hence it is zero by a 
modification of Liouville's theorem. Hence the left and right sides of (1.14) 
are identically zero, and this determines g+{a) and f-{a) simultaneoush^; for 
instance 

n 1f;^ . . . 2ik sin dp 1 1 

The growths are studied, for the above purposes, by noting, for instance 
in (1.12), that as a — ^oo in the lower half plane the resulting exponential decay 
makes the integrand neghgible except in the neighborhood of Xq = 0. Hence 
if we assume /(rco) '^ xl~^ near Xq = 0, we find 



S. N. KARP (S61) 415 

f-{a) '^ / xl~^ exp {—iaXo] dxo ^^ constant a"', 
(1.16) -^^ 

a— >oo with ^m{a) < 0. 

Here, also, we require p > 0, for convergence near the origin. 

(2) We now ^vish to emphasize a few features of the method of separation 
of variables. In this method a linear partial differential equation is written in 
a certain separable coordinate system ^, ^; then there exist solutions in the form, 

for various values of the separation parameter a. The functions fa , Qa arise 
from certain ordinary differential equations, of second order, and are combina- 
tions of their solutions, chosen to meet certain conditions of the problem in 
question. A more general solution is then 



V= f A(a)Ui)gM da 



where c is a suitable contour in the complex plane. When the contour encloses 
a sequence «„ of poles (only) of the integrand, one may also write 

F = E Aj^XOg.Sv), 

a Fourier series representation. Now the classical application of this method 
to boundary value problems envisions a boundary identical with the curve 
^ = ^0 , say. The solution of a boundary value problem is written in the form 
u = Uo -\- V where Uo is the incident field, and where the form of V has been 
given above. Then V has to adopt certain boundary values on ^ = ^o for all 
7}, these values typically serving to cancel those of ?io ? at ^ = ^o • Thus one 
has, for all r] in the region of interest, 

(2.1) = Firj) + f Aia)U{^o)gM da 

'' c 

with F{ri) given. This is an equation for the 'Tourier coefficient" A{a), and 
may be inverted by the use of a suitable 'Tourier" transform theorem. The 
existence of such generalized transform theorems has been discussed by Titch- 
marsh [7] and others. 

However, one may consider cases in which the coordinate system chosen 
does not conform to the boundary, i.e. the boundary in question is only part of 
the curve ^ = ^o • This means in other words that one represents the solution 
in terms of the eigen-f unctions of a simpler and different problem. In such a 
coordinate system equation (2.1) mil still hold in the range of interest in t], i.e., 
on the obstacle or boundary. But for ^ = ^o , and in other ranges of -q, different 
conditions may be derived and ^vill hold. These may result for instance from 
a requirement of regularity for u in the exterior of the obstacle. In this manner 



416 (S62) ELECTROMAGNETIC WAVES 

or in other ways, a mixed problem will arise, expressible in a pair of integral 
equations, for example 

(2.2a) F{ri) = I Aia)U^o)gM da 

(2.2b) (?(„) = f A{a)m,)gM da 

where ^(77) is given for 7) on the obstacle or boundary, and Girj) is given for rj 
off the boundary. Possibly also, the point set (^ = ^0 , ^ off the obstacle) may 
be further subdivided into various regions on which diverse conditions hold. 
In the present case the transform theorem alone is not directly applicable, for 
the value of the integral in (2.2a) say, is only given in part of the range of -q on 
the boundary. 

We wish to observe here that such a two part problem is often solvable by 
the intervention of function-theoretic techniques, and that if the boundary value 
problem for the partial differential equations were instead formulated as an 
integral equation with Green's function, it Avould give rise under these circum- 
stances to a ''Wiener-Hopf" problem in the extended sense, in terms of the 
transform relating to the coordinate system in which the problem is two part. 
This is in accordance Avith footnote 1, but is 7iot restricted to Cartesian coordi- 
nates. 

On the other hand, of course, if the problem is of the classical type first 
discussed, (and thus solvable by the use of transforms alone), then the corre- 
sponding Green's function integral equation is also directly reducible, hy use of 
the transform in question, to a simple equation between transforms, which can 
be solved purely algebraically. 

(3) We wish to illustrate the above remarks in concrete cases. Our first 
illustration relates to the problem of section 1, Equations (1.1) to (1.6) 

A solution by separation of variables which is small at 2/ = =t co , or alter- 
natively, corresponds to waves diverging from the positive x axis is, in Cartesian 
coordinates, 

exp {iax}-exp {i{k^ — af^^ \y\}' 

Hence we write, in this inappropriate coordinate system, 

(3.1) u, = =t| f ^P{a) exp {i[ax + {k' - aY' \ y |]! da 

where the plus or minus sign refers to ?/ > or ?/ < respectively. Then dUi/dy 
is continuous at ?/ = 0, while Ui may be discontinuous. Using (1.6), we find; 
with u = Uq -{- Ui 

(3.2a) / xp(a) exp {iax} da = 0, for a: < 

<J —CO 



S. N. KARP (S63) 417 

ik sin do exp lik(x cos ^o)} +9 / ^{oi){k^ — a)^^'^ exp [iax] da = 0, 
(3.2b) '^•^- 

for X > Oy 

cf. (2.2a), (2.2b). We now transform the integrals in these equations into 
contour integrals by adding infinite semicircles. The contributions of the latter 
are to be vanishingly small. In (3.2a), for instance, this is the case when the 
semicircle is in the lower half-plane, (in virtue of the decay of the exponential 
there for x < 0), provided yj/{a) is merely of algebraic growth at 00 in that half- 
plane. The upper half -plane is suitable for equation (3.2b). Expressing the 
excitation in suitable form for x > 0, by use of the Fourier theorem, we find 

(3.3a) / \p{a} exp {iax} da = 0, 

(3.3b) I (i Ha)(k^ - aT^ + ^HZllJ exp U«.! d. = 0, 

where Ci , Cg are the semicircles in the lower and upper half-planes, which have 
been referred to above. By Cauchy's theorem, (3.3a) would be fulfilled if 
\p(a) is regular in the closed lower half -plane. Define 

2\l/2 I '^ Sm Uq 



(3.4) ,(„).. ^(„)(,^_„r^+^_^^^^^^. 

Then (3.3b) would be fulfilled if 4>{a) is regular in the closed upper half-plane. 
Thus in (3.4) we may write <!) = (i>+ , \p = \p- , and we are in the position given 
by (1.13), with 4> corresponding to g, \f/ corresponding to/. We see here the role 
played by the algebraic groAvth and the region of regularity of \l/{a) and </)(q;). 

We wish to add here a remark on uniqueness. Due to the integrabiUty at 
the origin required by the methods of section (1), it was required there that 

g{x) - {-xy-\ fixo) - xr\ q>0, p > 0, 

in the notation of that section. This affected the growth of g+{a), f-{a), and 
ensured the vanishing of the entire function E{a). However, the procedure of 
section 1 is not obligatory. From the present heuristic standpoint, after manipu- 
lating (3.4) in the manner of section (1), we find, cf. (1.14) 



E(a) 



<l)+{a) /bsin dp f 1 1 

{k + «)'"' a- k cos ^0 \{k + a)'"- {k-\-k cos dof'^ 



fe sm ^0 _\- I ( \(h — ^l/2 

{a-k COS do)ik + k cos doY'' ^ 2 '^-^"^^^ ""^ ' 



418 (S64) ELECTROMAGNETIC WAVES 

but E(a) need not be zero, for certain growths of ^-(a), 0+(a). Then we have 



2ik sin dp i_ E(a) 

{a- k cos eo){k - ay'\k + k cos ^o)'"" "^ 2 (A: - a)' 



/Q cN f ( \ zz/c sm c/q ^ 

(d.5) ^P_{ol) = 77 7, _^ ^ ^n. ,M/2/7. . 7. ^^„ /)^l/2 + 



The first term corresponds to (1.15). Take for example (t)+(a) '^ a'^. Since 
</) corresponds to g of section (i), this imphes q{x) ^ ( — x)~^^^ near the origin, 
so that it would have been excluded. The corresponding E{a) is then a constant 
in (3.5), so that a term of the form (/c — a)"^^^ is added to i/'-(a), or a term of 
the form 

r exp {i\ax + (A:' - a^^ y \\) ^ 

signum 2/ • / ^ ^ 77;^ tttt^ ^^-^ c^a 

J-00 (/c — q;) 

is added to Ux . When 2/ = this is zero for negative x and a multiple 
of exp \ikx\lx''^ for positive a:, while its derivative with respect to y vanishes 
for 2/ = 0, a; > 0. Thus it corresponds to the addition of the solution 

U^nQ^r) cos (^/2), 

which does not affect the conditions of the problem. If for example E{a) = a, 
then we can interpret the contribution operationally as bjdx of the above homo- 
geneous singular solution. Also, the use of a suitable polynomial as E{oi), will 
produce multiples of the derivatives with respect to x of the original solution u. 
All such solutions are clearly admissible, as has been pointed out by Bouwkamp 
[8], so that for uniqueness it is necessary to specify the behaviour of the solution 
at the origin also. 

(4) From the present standpoint, we see that the Fourier transform solu- 
tion of the integral equation (1.10) is parallel to the representation of the solution 
of equations (1.1) to (1.6) in Cartesian coordinates (x, y). Here the cur^^e 
^ = ^0 cf. section 2, especially equations (2.2a) and (2.2b) is the line ?/ = 0; 
only part of this line, i.e. x > 0, is the boundary in question, i.e. the diffracting 
screen, so that we have a two-part problem. A more suitable coordinate system 
would be the system of polar coordinates r, ^, where the upper and lower surfaces 
of the screen are represented by ^ = 0, and B = 27r, respectively. In terms of 
this coordinate system, our boundary value problem is a one part problem of 
classical type. The eigen-functions are 



■^'(^r::}' 



and a more general solution is, say, 

(4.1) Ur = / J,{}zr)\A{y) cos vQ + B{y) sin vB\ dv. 

The problem may be solved directly in this manner, without the intervention 



S. N. KARP (S65) 419 

of function theoretic techniques, when one is in possession of an inversion formula 
for (4.1). Such an inversion formula has been obtained by N. N. Lebedev, and 
employed by the latter and M. J. Kontorowich [9] in the solution of this problem. 
The formula reads 

(4.2) cf>{kr) = -i f^^ v4>iv)JXkr) dv, 

(4.3) 4>{v) = r <f>{kr)Hi'\kr)^. 

Jo r 

(Actually the transformation (4.3) was applied to the partial differential equa- 
tion after certain preparatory modifications relating to integrability.) 

When the integral equation (1.10) is considered ab initio there is nothing 
to suggest the use of any particular type of transform. From the present stand- 
point it is seen that while it is a Wiener-Hopf equation, as far as the Fourier 
transform is concerned, it may also be inverted directly by the use of the Bessel 
transform of (4.2) and (4.3). It is convenient to represent the kernel in the form 



constant 

X Xq 



/too 
X' cot 7r\Hi^\kxo)Jx{kx) d\ 



which may be obtained by replacing the series for Ho[k{(x — Xq)^ -\r (y — yo)^V^^] 
by a contour integral, after suitable differentiations. Equation (1.10) takes the 
form 

ikx sin ^o exp { ikx cos ^o } 

r°° \u(x )] r'°° 

= constant- / dxo / X^ cot Tr\Hx{xo)Jx{kx} d\ 

/ A A\ «^0 ^0 J — ico 

= constant- / X[X cot Tr\uOC)]Jx{kx) d\. 

J — ico 

Hence by (4.2) and (4.3), 

X cot 7rX2^(X) cc iks'm do / exp {ik{x cos do)}Hx{kx) dx. 
The latter integral is obtained from the formulae (cf. [10]), 

j^ /^(aO|^°^^ ^J dt = y^ (m arcsin h/a)^, a > h, (Re^ > -1 
We find 



(4.6) Xfl{X) oc exp {-^X7r/2) 



sin X^o 
cos ttX 



420 (S66) 



ELECTROMAGNETIC WAVES 



or 



(4.7) 



[u{Xo)] oc j^ 



exp {—i\T/2} 



sin \do r n \ j^ 
cos Xtt 



The result is readily evaluated by residues. 

(a) We now T^'isll to make a few remarks \sith respect to the problem of 
the diffraction of a plane wave bj^ a staggered arraj^ of semi-infinite planes, a 
problem solved by Carlson and Heins [2] by reducing it via Green's theorem to 
a Wiener-Hopf integral equation. The geometry is depicted in Figure 2. 





1 

f 


/ 

/ 






Direction of 
incident ^y 


/ 
/ 


waves ^^^ 


/ 
/ 


A-' 




conducting 


4 ^ 

Figure 2 




screens 


a-7r<ffo<a<^/2 



Separation of variables does not seem appropriate here at first sight, but in 
^new of section {2), we may expect to find that a formulation in this way should 
succeed in a manner parallel to the Green's function imegral equation. To see 
this we represent the perturbation due to each plate (in a Cartesian coordinate 
system whose origin is at the leading edge of that plate) in the manner of (3.1) 



(5.1) 
with 

We set 



lim = =t 



\ f fM exp mx^ + {k' - ^y I y^ !]} rf/3 



Xm = X — md cot a, Vm = y — "tna. 



(5.2) 



u = exp {ik{x cos do -\- y sin do)} -\- ^ ii. 



Then, in order to avoid a discontinuity in u at y = nd (off the plate), we require 
by (5.1) that -w^ = at ?/„ = for a;„ negative. Hence, fm{?) must be regular 
in a lower half-plane and of algebraic gro^^1:h. We make a periodicity assump- 
tion on the current density, as done by Carlson and Heins, ^dz., 



(5.3) 



UmiXrrd = Uq{x) exp {ikm{d cot a cos 6 -\- dsmd)}. 



S. N. KARP (S67) 421 

By (5.1) 

/m(iS) = J UmiXm) exp {-i(3Xm} dXm 



= / UmiXm) eW {—i^Xm} dXr, 

'JO 



or 

(5.4) fM = /o(/3) exp likm{d cot a cos 6 -{- d sm 6)} , 
The condition du/dy = on metal, may be written 

~ =0, for a:. >0. 

This yields, 

(5.5) = /" [ ^T!\'c°os\ + /o(/3)^(/5)] exp {ifix,} d0 = 0, for x, > 0, 
where K(l3) is obtained from (5.4) and (5.1) as 

K{0) = i{k' - HT' f: exp [ipikp -0q) + i\p\ d(k' - ^y^'} 

p= — CO 

(5.6) 

{k' - ^y sin d(k' - ^y 



cos d{k' - n - cos {kp - /3g) 

with q = d cot a, p = q cos ^ + c? sin 6. 

If we define Z)(/?) to be the integrand of (5.5), then D((3) is to be regular in 
an upper-plane and of algebraic growth. The situation is then as in Carlson 
and Heins [2]. The use of the function theoretic techniques is necessary and 
suffices. It may be added that similar considerations apply to the case of a 
pair of semi-infinite plates (cf. [3]), with similar results. 

(6) The examples hitherto discussed or mentioned deal mth infinite bound- 
aries and depend insofar as their Wiener-Hopf character is concerned, upon the 
use of Cartesian coordinates. This is the case in the work of Carrier, and 
essentially also in the treatment of the semi-infinite cylinder, by Sch^vinger and 
Levine [4].^ In the present section, a finite obstacle will be considered. We have 
found that the semi-infinite plane constitutes a two part problem in Cartesians, 
but a one part problem in polars. The problem of a ribbon (under the conditions 
of section 1), represented in cross-section by 

y = 0, < a; < 1, 



^The latter employ cylindrical coordinates r, 6, z, and the transform relates to the ''Car- 
tesian" z coordinate. 



422 (S68) ELECTROMAGNETIC WAVES 

is, however, a three part problem in Cartesians, and the integral equation which 
results is a finite Wiener-Hopf equation. But, in polar coordinates, the problem 
is a two part problem. We sketch here the essential considerations in the case 
of Laplace's equation, in a very simple case which yet exhibits the . essential 
features. We seek a function u such that it is regular together with its de- 
rivatives except on the ribbon. On the latter u may be discontinuous, but not 
du/dy. On the ribbon du/dy is to be zero. More complicated solutions may 
be represented as integrals of this one with respect to x. 
A solution by separation of variables is 

r~"[A{v) cos vd + B{y) sin vB] dv. 

-too 

(An incident field may be represented in this way by means of the IMellin trans- 
form theorem, and included in more complex problems.) Continuity of du/dy 
at ^ = and 6 = 2w, requires essentially that B(v) = —A(v) cot tv. Con- 
tinuity of u ioT y = 0, X > 1 leads to 

/ioo 
r-'Aiv) dv = 0, r > 1 
- too 

which in turn requires A (v) regular in a right half-plane Giev > 0. The condition 
du/dy = on the ribbon leads to 



(6.3) / r-'vB{v) dv = 0, 



r < 1 



so that vB{v) is regular for (Re v < 0, for reasons similar to those of section 
Thus we haAX 

r T^/ N -, A / \ cos TTV 

(6.4) [^B{v)]. = -A^{v)v 



sm irv 



Using the identities 



(6.5) 



^ = r(.)r(i - .) 



sm TTV 



= Ki+'Mi-'). 





cos TV \z 


/ \:^ / 


we find 






(6.6) 


l*'l-tel= 


-^^Wr(i + .)' 



the left side being regular in the closed left half-plane and the right side in the 
closed right half. An entire function is thus defined, whose growth we study. 



S. N. KARP 



(S69) 423 



The gamma function quotients may be estimated by Stirling's theorem which 
leads to the formulae, 



r(i - .') 



(-")■ 



r(i + ") 

m + v) 



•1/2 



Hence, if in our case we let vB{v) have a growth of the order { — vY^^y and A{v) ^^ 
v~^^^, the entire function is a constant and this determines A{v) and B{v). 
There is a relation between the growth of these functions and the behaviour 
of u and d/dy near r = 1. We have 



(6.7) 



Jo 



ur 



■^ dr, 



(6.8) 



vB{v) = £ 



du 
dy 



r' dr. 



For large v with positive real part, the crucial part of the integral in (6.7) comes 
from r = 1. Say u behaves like (1 — r)~^ in that neighborhood. Then Aiv) '^ 
j; /-'{I - r)'-^"' dr = rWr(l - /3)/(l - /3 + ^). A similar result holds 
with (6.8), after a change of variable r = 1/p. 



^-0 




e=s. 






More compHcated geometries of this type are readily constructed. Such 
problems can admittedly be solved by other means. The more interesting 
problem of the wave equation and this geometry is still unsolved, due to the 
complex nature of the normalization of the Bessel functions as functions of v. 

(7) Another problem which can be treated in this manner is the problem 
of a truncated cone. The geometry is indicated in Figure 3. 






424 (S70) ELECTROMAGNETIC WAVES 

One may ask, for instance, for the electrostatic charge distribution on such 
an obstacle. We have then l^u = 0. Introducing the customary spherical co- 
ordinates and their product solutions, we write 

/» <r + t 00 

(7.1) . u = \ r"''AWP,_i(+cos e)dv, < 6 < do , 

nlJ+ico 

(7.2) u = / r-'Bip)P,_,{- cos 6) dv, do < d < tt, 

•J cT—ico 

these representations being designed to avoid the singularities of the Legendre 
function. We require u continuous with its derivatives except on the conical 
cup, where du/dd is discontinuous. On the cup, u = Uq , a. constant. (Problems 
involving oblique fields lead instead to sums of the form u = ^ u^ exp {irrut)}, 
■v\ath functions P^-i (cos 6) occurring in u^ •) Continuity of u at 6 = Oq imphes 

(7.3) ^(^)P,_i(+cos ^o) = 5(.)P,_i(-cos ^o), 
while continuity of du/dd for 6 = do with r > 1 leads to 

r\(T+ i CO ( /IT) 



e=ea 



(7.4) 

dv = 0, r > 1, 



-BW^(-cosS) 

or, defining (/)+(i') and utilizing (7.3) and the Wronskian of the pair of Legendre 
functions, 

<i>^{v) = ^(^)P:_i(+cos ^o) - 5(^^)P:_i(-cos ^o) 

,_ ^s ^ „ , . sin j'TT . Af \ sin vir 

(7.5) = const. B{v) - — r-, — - = const. A{v) - — z tt, 

^ ^ P,_i(+cos do) P,_i(-cos 6*0)' 

Regular for (Siev > o-, 
Also the condition u(r, do) = Uo ior < r < 1 leads to the relation, 

(7.6) f r-'\A(v)P,_,{cos do) -"^Idv = 0, r < 1 

when Uo is expressed as a Melhn transform in the range of interest. The inte- 
grand must hence be regular in a left hand plane Giev < a and if we call it g- (v) we 
have, using (7.5), 

(7.7) const. ^.(.) P^-('^o^^°)P>^.(-''°^^°) _ «2 = ,_(,). 

smirv V 

(The analysis has been completed, and will be detailed elsewhere. We 



s. N. KARP (871)425 

content ourselves here with brief indications of its nature, together with the 
exposition of an interesting special case.) We introduce the subscripts — J + 
II instead oi v — 1; then Pi/2+n (cos 6) is an entire function of /x, and its zeroes 
in the ^ plane are real, and are ultimately in arithmetic progression for large 
positive fi. Also if fi is a zero so is —fi. These facts permit the factorization of 
the coefficient of (p+(v) in (7.7) in the customary form K+(p)/K_(v), as well 
as an estimate of the growths of these factors by comparison with the gamma 
function. The introduction of a suitable exponential factor into K+{v) and 
K-(v) results in algebraic growth. The analysis in general follows the more 
perspicuous lines of the special case ^o = 7r/2, which represents a disc. For 
this special case, which we treat herewith, we have 

1/2 

P.-i(O) = 



r(i/2 + ^/2)r(i -v/2y 

expressing sin irv in terms of v/2 and using (6.6), (7.7) becomes 

^^•^^ ^^^^) •r(i/2 + ./2)r(i-./2) - 7 = ^-(^)- 

This can be recast in the form 

r(i - v/2) Up [ r(i - vi2) r(i) 1 
^-^ ^ r(i/2 - vl2) "^ V \r(i/2 - vl2) r(i/2)/ 
(7.9) 

^ -^or(i) (^.(^)r(^/2) 
^r(i/2) ^ r(i/2 + z./2)- 

If we now^ require in equation (7.1) o- < 2, then an entire function is defined. 
This amounts to an assumption on the behaviour of u at infinity. If <^+(^) '^ 
/ with i8 < J, and g-{y) ^^ v'^ with a < —^, the entire function is zero, by 
analysis like that of section 6, the significance of such assumptions having been 
discussed there. <t)+(v), A(v) and B(v) are now determined. This result agrees 
with one obtained by Titchmarsh [11], who started out with the equations 
(obtainable from cyhndrical coordinates), 






fiu)Jo(pu) du = Uq , < p < 1, 



f 



uf{u)Jo(pu) du = 0, p > 1. 



but eventually employed Mellin transforms via a representation of Jo(pu). 
His treatment of these equations has been improved by A. E. Heins [12]. 

A similar treatment of the disc problem for the wave equation case suggests 
itself, subject to the difficulties mentioned in section (6). 



426 (s72) electromagnetic waves 

Bibliography 

1. J. S. Schwinger, Fourier transform solution of integral equations, M.I.T. Radiation Labora- 

tory Report. 

2. J. F. Carlson and A. E. Heins, The reflection of an electromagnetic plane wave by an infinite 

set of plates, Part I, Quarterly of Applied Mathematics, Volume IV, 1946, pp. 313- 
330. Part II, Quarterly of Applied Mathematics, Volume V, 1947, pp. 82-89. 

3. A. E. Heins, The radiation and transmission properties of a pair of semi-nfinite parallel 

plates, Part I, Quarterly of Applied Mathematics, Volume VI, 1948, pp. 157-167. 
Part II, Quarterly of Applied Mathematics, Volume VI, 1948, pp. 215-220. 

4. H. Levine and J. S. Schwinger, On the radiation of sound from an unflanged circular pipe. 

Physical Review, Volume 73, 1948, pp. 383-406. 

5. G. Carrier, A generalization of the Wiener-Hopf technique, Quarterly of Applied Mathe- 

matics, Volume 7, 1949, p. 105. 

6. E. T. Copson, On an integral equation arising in the theory of diffraction, Quarterly Journal of 

Mathematics, Oxford Series, Volume 17, 1946, pp. 19-34. 

7. E. C. Titchmarsh, Eigenfunction Expansions, Oxford, London, 1946. 

8. C. J. Bouwkamp, A note on singularities occurring at sharp edges in EM diffraction theory ^ 

Physica, Volume 12, October, 1946, pp, 467-74. 

9. M. J. Kontorowich and N. N. Lebedev, Uher eine methode zur losung einigen prohleme der 

heugung^s theorie, Journal of Physics, Moscow, Volume 1, 1939, pp. 229-241. 

10. G. N. Watson, Bessel Functions, Macmillan, New York, 1945, p. 405, equations (4) and (5). 

11. E. C. Titchmarsh, Theory of Fourier Integrals, Oxford, London, 1937, pp. 334-337. 

12. A. E. Heins, A note on a pair of dual integral equations, Bulletin of the American Mathe- 

matical Society, Volume 56, No. 2, 1950, p. 172. j 

13. A. E. Heins, Equation integrates, Sur les couples d' equations integrates. Comptes Rendus de \ 

I'Academie des Sciences, Paris, Volume 230, 1950, pp. 1732-1734. 

j 



J 



Asymptotic Solutions of a Differential Equation 
in the Theory of Microwave Propagation 



By RUDOLPH E. LANGER 
University of Wisconsin 



The purpose of this paper is to show that asymptotic formulas for the 
solutions of a differential equation that is central to the theory of microwave 
propagation may be readily derived from results that are available in the mathe- 
matical hterature. The motivation for this paper comes from one by C. L. 
Pekeris/ in which certain derivations of such formulas are made on a basis of 
power series methods. Although that approach is suggestive, it has its ques- 
tionable aspects. Aside from the fact that it takes matters of convergence and 
rigor largely on faith, it seems fundamentally ill adapted, because of its essen- 
tially local character, for use in a problem which concerns an infinite range of 
the variable. It seems worth noting, because of this, that many needed formulas, 
dependably established, were already available in directly usable forms. 

As to the organization of the paper, the formulation of the problem is 
briefly reviewed in section 1, and a criterion for distinction between configura- 
tions in which equation (1) must be differently dealt ^vith is given. In section 
2, the differential equation under conditions which apply to the ''leaky" modes 
is discussed; especial attention is given to a method for deducing solutions that 
are explicit to more terms than the leading one. In section 3 the discussion is 
centered finally upon the differential equation under conditions which apply to 
the ''transitional" modes. 

In both sections 2 and 3 a qualitative comparison of the existing formulas 
with their analogues as obtained by power series methods is made. However, 
no actual quantitative check is included, since no such check has been undertaken. 

1. Introduction 

An analysis of the normal modes in microwave propagation in an atmosphere 
in which the index of refraction varies only with the height, is referable [P. 1108- 

^C. L. Pekeris, Asymptotic solutions for the normal modes in the theory of microwave propa- 
gation, Journal of Applied Physics, Volume 17, 1946, pp. 1108-1124. This paper will be re- 
ferred to by the designation P. 

Paper presented at the June, 1950, Symposium on the Theory of Electromagnetic Waves, under 
the sponsorship of the Washington Square College of Arts and Sciences and the Institute for 
Mathematics and Mechanics of New York University and the Geophysical Research Directories 
of the Air Force Cambridge Research Laboratories. 

427 (S73) 



428 (S74) 



ELECTROMAGNETIC WAVES 



10] in large measure to a determination of the forms of the solutions of a boundary 
problem in which the differential equation is of the form 



(1) 



U" + fc^A + ymU = 0. 



The variable h denotes the height, and is therefore positive and unbounded. 
Accents, such as those on C7, indicate differentiations with respect to h. The co- 
efficient y{li) stands for the modified index of refraction, the variables having 
been changed from the natural ones to such as make the earth's surface flat. 
This function is thus also real and positive. The case to be especially considered 
is that in which it decreases over some initial range of h, as is indicated by the 
graph in Figure 1. A state of super-refraction, with consequent improved trans- 



y(f^r 



FiGUEE 1 



mission, then exists in the atmosphere's ground layer. The values h and A in 
the equation (1) are constants. Of these k is positive and fixed, and is assumed 
to be large. A, on the other hand, is to be regarded as an eigenvalue parameter, 
for characteristic values of which the differential equation has solutions that 
fulfill prescribed boundary conditions. In the paper P. these conditions are 
that U have the character of an outgoing wave when li is large, and that it 
vanish at /i = 0.^ 

When a value (complex) of A is given, the equation 



(2) 



A + y{h) = 0, 



clearly has no real root. However, if y{h) is definable over an appropriate region 
of the complex plane, there may be a complex root hi . If so, the equation (1) 
may be written in the form 

(3) (7- + k'lyQi) - y{h,)]U = 0, 

with the result that it appears as a differential equation with a turning point at 



^The present paper is not coextensive with the paper P. The latter goes into the determi- 
nation of the eigenvalues Am and the normalization of the corresponding solutions Um , whereas 
the present paper is restricted to the asymptotic solutions of the differential equation alone. 



RUDOLPH E. LANGER ( S75) 429 

hi . A turning point is one at which the coefficient of the large parameter k^ 
has a zero. The possibihty of expressing equation (1) in the form (3) depends, 
of course, upon y{h) being a function whose definition is extensible over a region 
which contains a root of the equation (2). We shall suppose that in the cases 
to be considered this is possible, specifically that y{h) is thus extensible analytic- 
ally over a region that is contiguous Avith the axis of reals. The forms which 
asymptotically solve the equation (1) relative to k depend upon the order of 
hi as a root of the equation (2), namely upon the order of the turning point. 

It is the nature of an asymptotic formula relative to an unbounded parameter 
to be precise only in a limiting sense, and therefore to give only an approximation 
for any finite value of the parameter. In the case of equation (1), therefore, an 
asymptotic solution relative to k gives an approximation. The adequacy of this 
approximation for any particular purpose depends upon k being sufficiently 
large. Just what sufficient largeness may be is influenced by other factors, as 
is particularly clear in the case of equation (1), for k^ enters this only in con- 
junction "\^dth the multiplier [A + y{h)]. Any degree of largeness of k^ can 
therefore be vitiated by a corresponding smallness of this multiplier. Some 
normahzation of this parameter is therefore called for. It may be made as 
follows: 

If the turning point at hi is of the order ti, so that y^'\hi) = ior 1 ^ j < n, 
and y^''\hi) 9^ 0, the product k^[A + y(h)] may be written in the form 

(h - hiT. 



^2 y'-'Khi) 
nl 


y{h) - y{hi) 


^y ik - KT 



Here the factor Qi — hiY is merely specific of the order of the turning point. 
It is, therefore, accounted for in the very design of the appropriate asymptotic 
solutions. The factor within brackets, which embodies the whole remaining 
dependence of the quantity upon h, has been constructed to have at hi the 
limiting value 1. It has thus been normalized to be of moderate magnitude 
near hi . The constant k^y^""^ {hi)/n\ is thus left to fill the role of the parameter. 
It is accordingly this, rather than k^ alone, which must be sufficiently large if 
adequate approximations by the asymptotic formulas are to be assured. 

These considerations are important in the analyses of the differential equa- 
tion below. In section 2 the equation is dealt Avith when the turning point hi 
is of the first order. This point, however, is variable, due to its dependence 
upon A, whose eigenvalues are only subsequently to be determined. The condi- 
tion for adequacy of the asymptotic solutions is in this case that k^y\hi) be 
sufficiently large, and since the adequacy must obtain uniformly as to A it is 
clear that the condition must be uniformly fulfilled. This operates to bar hi 
from some neighborhood of the point /zq of Figure 1, since at that point y'{h) is 
zero. The formulas of section 2 are thus dependable only for values of A for 



430 (S76) ELECTROMAGNETIC WAVES 

which /?i does not he ^\^thin that neighborhood. For the excluded values, which 
are those that are associated with the transitional modes, the differential equa- 
tion is then considered in section 3. The turning point is there located at the 
fixed position ho . It is assumed that y"{h^ 7^ 0; more precisely that l^y"{ho)/2 
is sufficiently large. 

2. The Turning Point of the First Order 

If A is any value for which equation (2) has a simple root /zi , the differential 
equation (1) is expressible in the form (3) and thus has a turning point of the 
first order at /^i . The asymptotic solutions of such equations are kno^ii,^ and 
are conveniently expressed in terms of the follo^ving variables: 

v{h) = [A + y{h)r% 
Hh) = [ vQi) dh, 

(4) 

e{h) = *"(fe)*-'(ft), 

Q = hp{h), 






h 

dh. 



The functions 

(5) zM = mWH[%{u), j = 1, 2, 

in which the symbols Hi/l stand for the Bessel functions (Hankel functions) 
usually so denoted,^ are solutions of the differential equation 

(6) z'' + [kV(h) - d{h)]z = 0. 

Although the formula (4) for ^(/i) seems to assign a singularity to this function 
at /zi , it is found that this singularity is removable, and that ^(/ii) is then 
different from zero. The function d{h) is thus bounded in a region about h^ . 
Since the given differential equation (1) differs from (6) only to the extent of 



3R. E. Langer, On the asymptotic solutions of differential equations, with an application to 
the Bessel functions of large complex order, Transactions of the American Mathematical Society, 
Volume 34, 1932, pp. 447-480. This paper will be referred to by the designation Li . 

*G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge Uni- 
versity Press, 1944, pp. 73-75. 



M^ 



RUDOLPH E. LANGER (S77) 431 

this coefficient 6{h), it can be shown under suitable hypotheses that asymptotic 
solutions of (1) are obtainable from the known solutions (5) of equation (6). 

We shall make this more precise, and in doing so shall adjust the formulations 
so as to permit A to appear not as a fixed but as a variable parameter. It is 
•essential to do that if the formulas sought are to be usable in adjusting the 
solutions to boundary conditions. For, the eigenvalues of A for which the 
solutions fit given boundary conditions are usually unknown to begin with, and 
•can be determined only reflexively from the forms which the solutions are found 
to have. For the discussion in this section k^y'Qi^) must remain uniformly large. 
We shall therefore suppose that the range of A is a closed region of the complex 
plane for each point of which the equation (2) has a root h^ at which k^ \ y'(hi) \ 
exceeds some suitably large constant. As to this region, we shall suppose also 
that it includes no points of the axis of reals.^ The function y(h), being an index 
of refraction, is positive for positive h. We shall suppose that its definition is 
analytically extensible over some region of the complex plane that is contiguous 
with the axis of reals and includes all those values of hi which correspond to the 
admitted values of A. 

Under these hypotheses the left-hand member of the equation (2) has a 
square root whose real part is positive and bounded from zero when h is positive. 
We shall understand that (p(h) in (4) designates that root. We shall assume 
that the path of integration of the function ^(h) called for in (4) extends from 
hi to some point of the axis of reals and then is taken along this axis. The real 
parts of $(/i) and u are then increasing functions of h that become infinite with 
h. For large values of h, therefore, u is of large modulus and in the region 

-- < argw < -. 

We shall suppose, finally, that y{h) is such that the integral 



(7) f 






dh 



is convergent. A wide class of cases fulfill this condition, in particular any one 
in which y{h) has the character of either ch" or ce'^ for all large positive h; c 
and V being constants, with v "^ 0. 

The equation (1) then has [Lj , p. 459] a pair of solutions which maintain 
the forms 

^0.1 = r^ V'We^'"[i + oik-')i 

C/o.2 = A;-^V''We~'"[l + Oik-')], 



^This includes the case of the leaky modes. The discussion could easily be made to 
include also real values of A. That would, however, require some additional hypotheses and a 
consideration of various cases in some of which the range of h would have to be restricted to 
exclude other turning points. 



432 (S78) ELECTROMAGNETIC WAVES 

for large h, and are of the forms 

Uo.i = zM + 0(0, j = 1, 2, 

when h is such that | w | is moderate or small. Upon taking U* and L'** as 
suitable constant multiples of these, we therefore have the fact that there are 
solutions of equation (1) which have the forms 

U* = {2/7ry''Q-''V'-'''''''[l + 0{k-% 
(8) 

C/** = {2/7ry''Q-'''e-'''^'''''''[l + Oik-')], 

for large h, and are described by the formulas 

U* = [u/Qr'H\)l{u) + 0{k-"'), 
(9) 

for moderate or small values of u. The solution C7** is clearly the one which 
has the form of an outgoing wave. 

The formulas (8) and (9) are explicit only to the extent of their leading 
terms. If more precise forms are needed, such may be found by the folloT^-ing 
procedure.^ In terms of the functions (5), let cciQi) and oi^Qi) be defined thus 

(10) «,w = [i + ^},w + ^«;(/o, i = 1, 2, 

T^dth tentatively undetermined coeflScients a{h) and /3(/0- By differentiation and 
subsequent elimination of z'/ through the use of equation (6), it is foimd that 

(11) <^m = [-^'fi + ^4^}, + [i + ^}; , 

and a repetition of the process yields a corresponding formula for co,-' in terms 
of 2 J and z'i . Thus 



co;' + kVQi)o^i = [^0 + ~i\z^ + 1^^ 



^R. E. Langer, The asymptotic solutions of ordinary linear differential equations of the second 
order, with special reference to a turning point, Transactions of the American Mathematical 
Society, Volume 67, 1949, pp. 461-490. This paper will be referred to by the designation La . 
The method was also outlined in R. E. Langer, On the connection formulae and the solutions of 
the wave eqvxition, Physical Review, Volume 51, 1937, pp. 669-676. 



RUDOLPH E. LANGER (S79) 433 

with 

5j = a" + ^a + 26^' + 0'/3, 

By the choice of ^{K), and then of aQi), we may make >So = and ^2 = 0. 
The formulas for this are 



''('') = 2^)1 



e{h) 



dh, 



2<p{h) A, <pih) 
(12) 

a{h) = -| [/3'(/i) + £ ^(/i)/3(/i) rf/ij, 

and it is to be observed that, although /?(/?) appears to be singular at h^ , this 
singularity is removable. With the determinations (12) we have, therefore, 

(13) co;' + kV(h)c^i = (s,/k')z,- , j = 1, 2. 

Equations (10) and (11) are solvable for Zj in terms of coj and co' , and yield 

with 

^ _ , . 2a + ^' + <py , a' + a^' - a'l3 - 60" 
^ - ^ + k' + k' 

The result of substituting this evaluation into equation (13) is the relation 

(14) «" + i%o>' + {fciA + y{h)] + ^ [1 + 4^]}" = °' 

with coy in the place of co. Since this follows for both coi and C02 , it has been 
found that the functions (10), (12), are solutions of the differential equation 
(14). This equation differs from the given equation (1) only by a coefficient of 
the order of k~^ for co' and one of the mder of k~^ for co. From this it can be 
shown [L2] that there exist solutions of (1) which are subject to the descriptions 

[/,. = a>,W[l + 0{k-')], j = 1, 2, 

when h is large, and 

U, = o>,(.h) + 0{k-'), 



434 (S80) ELECTROMAGNETIC WAVES 

when u is moderate or small. On multiplying these by k~^^^ and using the ab- 
breviations 

(15) Z,(h) = [u/Qrm%(u), j = 1, 2, 

the conclusion may be stated as follows: There exist solutions of the equation 
(1) which maintain the forms 



U* = 
(16) 



1 + 



^]zM + ^zm][i + 0{k-')], 



1 + ^]z2{h) + ^zm}[i + o{k-')], 



when h is large, and are given by the formulas 



f/* = [i + ^]z.(h) + ^zm + oik-'n. 



(17) 

TT** = I 1 4- ^ Iz.rw 4- ^ 

k 



V** = [i + ^]z,{h) + ^zm + oik-^n 



when u is moderate or small. These are the more explicit versions of the formulas 
(8) and (9).'"' 



'^This method may be used [L2 , 472-3] to derive formal solutions that are expUcit to an\' 
prescribed powers of l//c. 

^The formula which was derived by Pekeris (see footnote 1) as the counterpart of the 
second formula (17), is [P, 20] 

U^ = ^5 H[%(u) + 2^^ H%,(t^) - ^1^ Hi%{u) + Q-''%)0{k'n, 

with coefl&cients A and B which are constants expressible in terms of the values of y{h) and its 
first four derivatives at hi . This can be given a form much more similar to that of U**. For 
(see footnote 4) it can be shown that 



5/6 _ 



Ql/2 J^ -2/3\(^J — 1 



u'" 



^^» = {i+i©>-^^ 



Thus the formula can be written alternatively 

?7„ = [1 + ^-^§\zm + ^zm + Q''\h)o{k-''\ 

with 



RUDOLPH E. LANGER (S81) 435 

3. The Turning Point of the Second Order 

When A is on a range of values for which the root hi of equation (2) lies 
too near the point ho of Figure 1, so that k^y'Qii) is inadequately large, the 
analysis must be cast along different lines. With c as a constant subsequently to 
be determined, let 

h = X -{■ (c/k). 

From Taylor's formula we have then 

y(h) = y(,x) + I y'{x) + f, y"[x + |), 

with Ci some value between and c. We may, therefore, write 

k'[A + y{h)] = k'lyix) - y{ho)] + k[k{A + y{ho)} + cy'ix)] + cY'\x + |}, 

and thus, on setting 

X = ki, 

Xo(a:) = y{x) - yQio), 
(18) 

Xiix) = i[k{A -h y(ho)} + cy'(x)], 

X,{x) = -cV'{a; + fe//b)}, 
we may write (1) in the form 

(19) ^ - [X'Xlix) + \X,{x) + X,{x)]U = 0. 



There is thus no qualitative conflict between the two results. No quantitative check upon them 
has been made. 

The method by which the results of the present paper were derived is completely different 
from that of P. The latter is based upon the power series for y{h) about the point hi . Through 
operations upon this series, the coefficients of the differential equation for Q^'^ C/ as a function 
of u are found in terms of power series in u. By retaining only the leading terms of these series 
an approximate differential equation is found. Even leaving aside the question of convergence 
of the various series which are thus brought into play, it will be clear that such a deduction 
bases itself entirely upon the character of y{h) at the point hi , even though the identification of 
the solution which is an outgoing wave must be made by reference to arbitrarily large values 
of ^. 



436 (S82) ELECTROMAGNETIC WAVES 

It appears therefore as a differential equation ^dth a turning point of the second 
order at ho , since at this point the coefficient Xo(x) of the highest power of 
the large parameter X has a zero of that order. The function Xifxj, although 
it involves the constant h, is not large; for k appears in it only with, the multiplier 
{A + y{ho)}, which is evidently small since it is expressible as the integral of 
i/ih) over the range from hi to ho where it is smalL 

A differential equation (19) is defined^ to be in the normal form if its co- 
efl&cients fulfill the relation 



^I 



3X^X( - 2X^'Xi =0. 

L Jh=ho 

In the case at hand this may be assured by assigning an appropriate value to 
the constant c, namely by setting 

2kU + y(h„)W"(ho) 
' q[y"{ho)f 

Under suitable hypotheses the forms of the solutions of the differential equation 
are then known. For use in expressing them we shall set 

-kilA + y(ho)} 



2[2y'\ho)] 



1/2 



(20) 



„fe) = X^ , 2^Xo(x) 
"^^^ XoW + J^.Xo(x) dx' 



<p(x) = 2Xo(x) + ^, 



^ = X / (p{x) dx. 



The singularity of r](x) at ho is removable, because of the normafitj^ of the 
differential equation, and when it is removed rjQio) = 0. If the function y{h) 
is such that Xo(x) is bounded from zero except in the immediate neighborhood 
of ho , and if certain other conditions of a somewhat similar nature are fulfilled 
[Lg, 93, 101], the following is assured [Lg, 105, 106]: equation (1) has solutions 
which are of the forms 

U,,, = \-''%-'''(x)re"'"'ll + 0(X-')], 
C^o.. = X-' V'(x)re-"'"ni + 0(X-')], 



3R. E. Langer, The asymptotic solutions of certain linear ordinary differential equations of 
the second order, Transactions of the American Mathematical Society, Volume 36, 1934, pp. 
90-106. This paper will be referred to by the designation L3 . 



RUDOLPH E. LANGER (S83) 437 

when h is large, and such that 



1/2 
TT 



+ '^^7i'lrT^ m,.-Uq} + 0(x" log X), 

1/4 -1/2/TN.' -^TT 



r(i + <r) 



1/2 



+ rj- ^) ^^--1/^(0/ + 0{\-' log X), 

when I ^ 1 is moderate or small. The symbols M^.^^j are used here to denote the 
confluent hypergeometric functions that are customarily so designated.^ 

With t/* and f/** taken as suitable constant multiples of the f/o,/ , and 
with the variables Q, u and r defined as 

Q = k[y{x) - y{K)r\ 
(21) u= f Qdx, 

T = / 7] dx, 

the conclusion may be restated as follows: the differential equation (1) has 
solutions which for large values of h maintain the forms 

(22) 

Tj** (2iM + r)' ,„-(i/2),. nci,-'^\-\ 

and are given for moderate or small values of u by the formulas 
C7* = (0-fP{j^M„U2.«+r) 



+ r(j"+a) ^^"•-■/'(^m + t) ^ + 0(/c-=^* log A;), 



i^E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 3rd ed., Cambridge 
University Press, 1920, p. 337. 



438 (S84) ELECTROMAGNETIC WAVES 

(•\-l/2< c\ 1/2 
- f j | r(i !. ^) M..,/4(2m + r) 

(23) 



+ r(f - ^) ^— i/*(2m + t)| + 0(/b-^/^ log h). 



Of these solutions, C/** is evidently the one which has the form of an outgoing 



wave." 



"The paper P gives in the place of the second formula (23) the following one: 

f 9 1/2 _i/2 1 



r(J - k) — /— ™vv. . p(i _ ^^ 
with K differing from <r, thus 

t[7?/"'(/i,) - 9?/"(feo)] 

The most conspicuous discrepancy between this and the second formula (23) is in the sign of 
the term in Mk,i/4 . One suspects a typographical error in this. There are, of course, also a 
number of other differences, all of which however would approach zero if h were to become 
infinite. 



Criteria for Discrete Spectra 



By K. 0. FRIEDRICHS 

New York University 

In discussing the nature of the spectra of differential operators I shall 
confine myself primarily to ordinary differential operators of the second order. 
Later on I shall indicate how this discussion can be extended to partial differ- 
ential operators. In fact, one of the main advantages of the approach I shall 
employ is that it is not restricted to ordinary differential operators but can to 
a large extent be employed for partial differential operators. 

The differential operator L we shall consider is of the second order, and 
seff-adjoint; it acts on functions (f){x) which are defined in an interval ^ and 
subjected to various admissibility conditions. One may then ask for eigen- 
functions 

(1) = u(\; x) 

of this operator. These eigenfunctions are certain solutions of the eigenvalue 
equation 

(2) L0 = \cf>; 

it is desired to expand arbitrary functions 0(x) in terms of these eigenfunctions. 
Since the differential operator L is of the second order, it should be possible to 
write the eigenvalue equation in the familiar form 

(3) £ vix) £ 0(a;) - q{x)cl>{x) + XT{x)cf>(x) = 0, 

in which three given continuous functions p(x), q{x), and r(x) appear. In order 
that equation (3) be equivalent to equation (2) we must define the operator L 
by 

(4) L = .-'(x)[-£p(x) + £g(.)]. 

The coefficients r{x) and p(x) are supposed to be positive in the interior 

Paper presented at the June, 1950, Symposium on the Theory of Electromagnetic Waves, under 
the sponsorship of the Washington Square College of Arts and Sciences and the Institute for 
Mathematics and Mechanics of New York University and the Geophysical Research Directories 
of the Air Force Cambridge Research Laboratories. 

439 (S85) 



440 (S86) ELECTROMAGNETIC WAVES 

of the interval ^. If the interval ^ is finite and if the functions r(x) and p(x) 
are continuous and positive at the endpoints and if also q(x) is continuous there, 
the operator L is said to be regular or of the Sturm-Liouville type. It is well 
known that for such regular operators a complete sequence of eigenvalues X 
and eigenfunctions u(\; x) exists provided that appropriate boundarv^ conditions 
are imposed at the endpoints. The sequence is caUed complete if arbitrarj^ 
functions (f)(x) of a certain class admit an expansion of the form 

(5) ct)(x) = J2 ci\u(\; x) 

with appropriate coefficients ax . 

We are here concerned with a singular operator L, i.e. with one for which 
not all regularity conditions mentioned above are satisfied. Thus, the operator 
L is singular, if the interval extends to infinity in one or both directions, or if 
one of the functions r{x), pix), r~^(x), p~^{x), q{x) does not approach a finite 
value at one endpoint at least. 

It is well known that there are singular operators L which do not possess 
a complete sequence of eigenvalues and eigenfunctions in a proper sense. Xever- 
theless, for these singular operators there exists an analogue to the expansion 
of arbitrary functions with respect to eigenfunctions. In this modified expan- 
sion, integration occurs instead of summation, invoMng "improper" eigen- 
functions, v(k; x), which depend continuously on the eigenvalue X in a certain 
set S of values X. Thus the expansion formula is of the form 

(6) cl>ix) = f hi\)v(\; x) d\-\- J2 cxuiX; x). 

J S X 

The set S in which the improper eigenfunctions v{x; X) are defined is called the 
continuous spectrum of the operator L; the values X occurring in the sum form 
the point spectrum. The functions u{\; x) will also be called ''proper" eigen- 
functions. 

While it is true that for certain singular operators L onl}^ an expansion of 
the type (6) is possible, there are nevertheless certain classes of singular operators 
for which an expansion of the type (5) with respect to proper eigenfunctions is 
possible in the same way as for regular operators. We then say that the operator 
L possesses a pure point spectrum; if the eigenvalues form a sequence tending 
to infinity we speak of a totally discrete spectrum. It is desirable to be able 
to determine from the nature of the coefficients p, q, and r whether or not the 
spectrum of L is totally discrete. The present discussion is prunarily concerned 
with criteria which enable one to do this. More generally, we shall be concerned 
with criteria which will enable one to decide whether or not the part of the 
spectrum which lies below a certain value X^ of X is discrete, i.e., consists of a 
finite number of point eigenvalues, so that the continuous spectrum S, if there 
is any, is confined to values X > X^ . 



K. O. FRIEDRICHS (S87) 441 

Before formulating such criteria in detail I should like to mention a typical 
problem of wave propagation in which one is naturally led to ask whether or 
not the spectrum of an operator L is totally or partially discrete. Let us assume 
that the propagation of a field quantity x = x(^j 2;, t) in a stratified medium is 
governed by the differential equation 

(7) ''(^)5?'' = toP(^)te^ + ai''(^)aJ'^' 

involving positive coefficients vix), r{x) and p{x) which depend on one co- 
ordinate, X, but not on the other, z. Let us ask whether or not there are waves 
of the form 

(8) x{x, z, t) = e""e-""<t,{.x), 

having a given frequency k and progressing in the ^-direction, with the velocity 
k/ji. The frequency k being given, the wave number 11 is to be determined. 
The amphtude 4){x) of such weaves evidently satisfies the differential equation 

^^^ fx ^^''^ ic ^^^) ~ /^'^W^W + k'p(x)cl^{x) = 0, 

which agrees with equation (3) if one sets 

(10) q(x) = -k%{x), X = -/, 

assuming the frequency k to be fixed. It is now particularly desirable to know 
whether or not there are unattenuated modes of propagation corresponding to 
proper eigenf unctions cf){x) = u{\; x) associated with discrete negative eigen- 
values X = — M^- III other words, one is interested in knowing whether or not 
the spectrum is discrete below the value X = and how many negative point 
eigenvalues there are. The criteria to be discussed will, in general, enable one 
to answer this question. 

A complete theory of the expansion of an arbitrary function with respect 
to proper or improper eigenfunctions of a regular or singular ordinary differential 
operator L was given by Weyl in 1909 [1]. Treatments of this problem, more 
or less differing from that by Weyl, partly less general, partly extending or 
simplif3dng it, were given by Stone [2], Friedrichs [3], Titchmarsh [4], and 
Kodaira [5]. Sufficient conditions for the discreteness of the spectrum, of a more 
or less special character, were given by these authors in the course of their in- 
vestigations. 

Before describing such criteria, the eigenvalue problem of the operator L 
should be formulated in a more precise manner. The functions 0(a;) should be 
quadratically integrable in the sense that 



ai) (0,0) = [ rix) \<p(x) \'dx < 

J a 



442 (S88) ELECTROMAGNETIC WAVES 

The manifold of all such functions ^^'ill be denoted by §. The operator L, given 
by (4), is not applicable on all functions in §, but only on those which are con- 
tinuously differentiable and such that the function p{x) 64 /dx also has a deriva- 
tive. We require that the function L<i){x) also belong to §. The manifold of all 
functions 0(.t) for which this is the case will be denoted by %^. In order that 
the operator become hypermaximal^ so that an expansion with respect to eigen- 
functions is defined in a unique way it may be necessar}^ to impose further 
conditions, boundary conditions, on the functions 0(a:) in g°. 

Weyl discovered the important fact that there are just two cases possible 
for each endpoint: either one of a certain linear manifold of boundarj^ conditions 
must be imposed, or, no boundary conditions should be imposed at all. If at 
each endpoint one of the admissible boundary conditions is imposed in the 
first case or none in the second case the operator L ^\dll become h\i)ermaximal. 
Weyl distinguished these two cases as limit circle and limit jooint cases, terms 
tliat were suggested by the particular approach that he employed. I shall not 
discuss here any details about this matter; I shall onh^ mention that it is natur- 
ally important to be able to deduce from the nature of the coefficients p, q, r, 
which of the two cases arises. Various criteria about this alternative were given 
by the authors mentioned. 

In the foUomng we assume that at each endpoint one of the permissible 
boundary conditions has been imposed if necessary.^ The manifold of functions 
0(x) in g° satisfying these conditions \\\\\ be denoted by g. The proper eigen- 
functions should then belong to the space g. This requirement implies, in 
particular, that the integral (11) should be finite for them. 

In case the spectrum of the operator L is totally discrete according to the 
definition given above, there exists a sequence of eigenvalues X which increase 
and tend to infinity, X t co . Every function 0(x) in § possesses an expansion 
of the form 

4>ix) = ^c^ui},] x). 

X 

For functions <i){x) in %, in addition, the expansion 
(12) L4>{x) = X >^w(X; x) 

X 

holds. Here the summation refers to the sequence of eigenvalues and con- 
vergence is understood to hold in the mean. The coefficients Cx are given in 
terms of the function 0(x) through simple integrals. The spaces ^, or g, are 



^Instead of this term, introduced by von Neumann, the term "self-adjoint" was employed 
by Stone. We have not used the latter term in order to avoid confusion with the property of 
"formal" adjointness of a differential operator. 

^For a convenient way of determining the appropriate boundary conditions, see Rellich [6]. 



K. O. FRIEDRICHS (S89) 443 

also the largest manifolds of functions for which respectively the first, or both, 
of these expansions hold. 

For the present purpose it is suitable to adopt the following definition of 
partial discreteness of the spectrum: We say the spectrum of the operator L is 
discrete below a value X of \ if to every value X' < X there exists at most a finite 
number of mutually orthogonal eigenfunctions Ux{x) associated with eigenvalues 
X < X' such that for every function in % which is orthogonal to the eigenfunctions 
ux = u{\; x), 



(13) 


{ux , 0) = 0, 


the inequality 




(14) 


(0, L4>) > \(cf>, cf>) 



holds. It then follows from the general theory that if the spectrum is discrete 
below every value X^ , it is totally discrete. 

After these preparations we may formulate various criteria. 

To this end it is convenient to introduce a function h(x) which is positive 
and of the form 



(15) h=\ [ -~ + C 

\ J v(x) 



p{x) 

in the neighborhood of each endpoint, x = x^ or x = x+ , o^ ^. The constants 
C = C- and C = C+ should be so chosen that at each endpoint h is either zero 
or infinite. In other words, the positive function h(x) should be so chosen that 
I dh/dx \ = 1/p near the end points and that it becomes either zero or infinite 
at the endpoints. 

We further introduce the quantity 

We maintain that the discrete character of the spectrum depends on the behavior 
of this quantity Z{x) at the endpoints x = x- and x = x+ . The first criterion is 

I. The spectrum of the operator L is totally discrete if 

(17) Z{x) —^^ as x -^ x^ and x —^ x+ . 

Certainly, this test is easily applied. Let us, in particular, take the case 
that r{x) = p{x) = 1, x_ = 0, a;+ = oo . Then h{x) = x, and the criterion re- 
duces to 

(17') q{x) + 2~2 -^°° as x -^ and x -^^ . 



444 (S90) ELECTROMAGNETIC WAVES 

This condition is certainly satisfied if q(x) — >oo as x ^oo while q(x) remains 
bounded at x = 0. That the spectrum is discrete under this condition is a very 
familiar fact, which plays a considerable role in spectral problems of quantum 
theor^^ It is to be noted that condition (17') admits also functions q(x) which 
are negative infinite as x — ^ 0. A singularity of g(x) at x = like that of a 
Coulomb potential, q{x) = —c/x, for example, does not spoil the totally dis- 
crete character of the spectrum. 

It may be mentioned that in principle it would be suflacient to formulate 
criteria under the condition p(x) = r(x) = 1, since this condition can be satisfied 
by introducing new dependent and independent variables instead of </> and x 
provided p(x) and r(x) possess second derivatives. However, when expressed in 
terms of the original coefficients p, q, and r, condition (17') becomes very com- 
plicated and, therefore, it seems preferable to use the rather simple condition (16). 

Relation (17) is not a necessary condition for totally discrete character, 
but it is almost a necessary condition, as seen from the following two criteria. 

II. The spectrum of the operator L is discrete below the value X = X^^ ^f 

(18) lim inf Z{x) > \ 
with reference to x ^ X- and x —^ x+ . 

III. The spectrum of the operator L is not discrete helow X!^' , if 

(19) limsupZ(:c) < X^ 

with reference to either x -^ x- or x -^ x+ , provided that Z(x) is bounded below. 

Consequently, if Z{x) approaches a definite hmit X^^ at one endpoint and 
has an inferior limit >X^^ at the other endpoint the spectrum is discrete below 
X^^ but not below any value X^ > X^^ . In other words, a non-discrete spectrum 
begins at X = X^^ . 

The conditions of these three criteria imply that the quantity Z{x) is 
bounded below. As a consequence, as can be sho^Am, also the spectrum of the 
operator L is bounded below. If, however, Z{x) approaches — oo at one end- 
point, the spectrum also extends to — co . This spectrum lasiy be a continuous 
spectrum, but strangely enough, the spectrum may even be discrete. We then 
speak of a discrete spectrum unbounded beloAV. More specificalty, we saj" that 
the spectrum of the operator L is discrete below X^ , unbounded below, if there is a 
sequence of eigenvalues X < X^ which decreases and approaches oo such that in- 
equality (14) holds for all functions in g which are orthogonal to all corresponding 
eigenf unctions, i.e. which satisfy condition (13). If this is the case for every 
value X^ , an increasing and decreasing sequence of eigenvalues exists such that 
the expansions (5) and (12) hold; we then say the spectrum is totally discrete 
unbounded below. We now state, slightly generalizing a criterion due to ReUich. 



K. O. FRIEDRIGHS (S91) 445 

IV. The spectrum of the operator L is discrete below X^ , unbounded below, if 
lim inf Z{x) > X^ at one endpoint, e.g. x = X- , while for the other endpoint, x = 
x+ , the relation 

(20) Z(x) — > — CO as X ^ x+ 

holds in such a way that 

Xo being any number such that Z(x) < for x = Xq . Of course, the roles of x+ 
and X- could be interchanged. (The vahdity of this criterion has so far been 
proved only if the functions p, q, r satisfy additional smoothness conditions; for 
such additional conditions in case p = r = 1, x+ = ^, see [4].) 

In case p(x) = r(x) = 1, q(x) continuous and negative, < re < oo, and 
x+ = CO , conditions (20) and (21) may be replaced by 

(20') q(x) —^— CO as a;— ^00 

and 

dx 



(210 / 

Jo 



q(x)] 



1/2 



<oo 



If, in addition, the function q(x) is defined continuously at x = 0, the spectrum 
is totally discrete unbounded below.^ If condition (20') is satisfied but not 
(21') the spectrum is continuous; this interesting statement is due to Titchmarsh 

[4]. 

As an example of the application of criterion IV we may take the case of 
spherical waves characterized, with reference to equation (9), by 

p{x) = x^, r(x) = 1, p{x) = x^y X- = 0, a;+ = 00, 

so that q{x) = -/cV, k ^ 0, by (10) and 

h{x) = x-\ Z{x) = -A;V + i- ■ 



Hence 



Z{x) -^ i as 
Z{x) — »— oo as 



^This remarkable fact was mentioned to me by Rellich. Addition in proofs: It is also 
stated in a recent paper by Sears and Titchmarsh [9] correcting erroneous statements made 
in [4]. 



446 (S92) ELECTROMAGNETIC WAVES 

and 

^= / •/72 4 "T77TT72 < °° 



dx 

(A;V - a:74)^ 



Consequently, the spectrum is discrete below X = 1/4, unbounded below. The 
nondiscrete character of the spectrum above X = 1/4 is due to the behavior at 
the origin. 

Various different criteria have been formulated in the literature; some of 
them give more information in the case that the inferior and superior limits of 
the function Z{x) at one endpoint differ from each other. I do not intend to 
mention here those fine points of difference. It is also possible to prove the 
validity of these criteria in many different ways. I shall indicate only one ap- 
proach which has the advantage that it does not rely on the fact that the differ- 
ential operator is an ordinary differential operator and which, therefore, to a 
large extent can be used for partial differential operators. 

This approach is based on the fact that hypermaximal operators with 
a totally discrete spectrum can be characterized by an abstract propert}^, which 
in most concrete cases is easily verified. This important property was dis- 
covered by Hilbert [7] in 1906 for the case of bounded operators, such as integral 
operators. Later on, Weyl [8] extended this property to operators \\ii]i a 
partially discrete spectrum and also made use of it for differential operators, 
which are not bounded, after having transformed them into bounded integral 
operators with the aid of a Green's function. 

It is, however, possible to utilize the ideas of Hilbert and ^Qy\ for differ- 
ential operators directly. In carrying this out one obtains at the same time 
upper and lower estimates for the number of discrete eigenvalues below an 
arbitrarily chosen value. On the other hand, this direct approach seems to be 
restricted to the case in which the quantity Z{x)j and hence the spectrum, is 
bounded below. 

For simplicity we restrict ourselves to the case in which a particular bound- 
ary condition is imposed. 

We consider the quadratic forms 



r{x)<f>\x) dx, 



and 



(22) {cl>G<f>) = £' [?'(^)(£ ^(^))' + ?W<^'W J dx, 

further the space § of all functions (f)(x) for which (0, 0) is finite and the space 
& of all differentiable functions <j>{x) for which (^G^) is finite and which tend 



K. O. FRIEDRICHS (S93) 447 

to zero as x approaches an endpoint at which the function h(x) vanishes, see 
(15). Then we state 

A. Suppose there are n functions 4>^^\x), ••• , <f>^''\x) in § su^h that the 
inequality 

(23) (^(?0) + i: (0, 0'")^ > X'(<^, 4,) 

holds for all functions <f){x) in ®. Then the spectrum of the operator L is discrete 
helow the value X' and there are at most n eigenvalues of L below \'. 

In case the inferior Hmit of the quantity Z{x) at the endpoints is greater 
than X', it is not very difficult to construct a finite number of functions 4>'^''\x) 
such that inequahty (23) holds. In this way, the vahdity of criterion II and 
hence of I, can be established. At the same time an upper estimate, n, of the 
number of eigenvalues X < X', is found. 

The proof of criterion III for non-discreteness of the spectrum can be given 
in connection with the following fact: 

B. Suppose there are m linearly independent functions \f/^^\x), • • • , \f/^"'\x)j 
in ® such that for every linear combination 

(24) 0(x) = E c,^"\x) 

which does not vanish identically , 

(25) <j,{x) ^ 0, 
the inequality 

(26) (06^0) < X^((^, 4>) 

holds. Then either there are at least m point eigenvalues below X^ , or the spectrum 
is not discrete below X^ . 

Under the conditions of criterion III it is not difficult to prove that there 
are arbitrarily many eigenvalues below a certain value X' < X^ . This fact 
imphes that the spectrum is not discrete below X^ . Thus the vahdity of criterion 
III can be established. 

Statement B evidently enables one to obtain a lower estimate, m, of the 
number of eigenvalues below the value X^ . We illustrate this fact with the 
rather trivial example in which p{x) = r{x) = 1 and q = — k iov < x < a, 
while g = Ofora<x<co. We simply take the functions yp^"^ {x) = sin {inrx/a)^ 
X < a, =0, X > a, for V = 1, • • • , m. For any linear combination 0(x) = 
2l7=i c^\f/^''\x) of these functions we evidently have 



448 (S94) ELECTROMAGNETIC WAVES 






m / 2 2 . „ 

^=1 \ ft 



2 



and hence 



From statement B we now infer that there are at least m negative eigenvalues if 

m < Ka/ir. 

If one intends to employ the criteria A and B in concrete cases, one must 
find appropriate functions (p^''\x) and \p^''\x). To do this right requires skill 
and experience since there do not seem to exist hard and fast rules that tell 
one how to find such functions, although a few results of experience could be 
formulated. 

These two criteria A and B have various ramifications. I only mention, 
as P. Lax has noticed, that the properties A and B are respectively equivalent 
with the converse properties B and A, which can be put to good use. Also there 
is a close connection between properties A and B in Courant's maximum mini- 
mum property and a corresponding minimum maximum property. 

It is rather evident that the criteria A and B can also be formulated \^'ith 
reference to partial differential operators; all that is needed is to define analogues 
of the forms (11, 22). As a consequence, the criteria I to IV can be carried 
over to the case of partial differential operators provided the functions p, g, r 
behave uniformly at each component of the boundary. Certainly it should be 
possible to relax even this condition. This extension to partial differential 
operators has, however, been carried out only in special cases. 



Bibliography 

1. Weyl, H., Uher gewohnliche Differ entialgleichungen mit singuldren Stellen, Gottinger Nach- 

richten von der Gesellschaft der Wissenschaften, 1909, pp. 37-64. Uber gewohnliche 
Differentialgleichungen mit Singular itdten, Mathematische Annalen, Volume 68, 1910, 
pp. 373-392. Ramifications, old and new, of the eigenvalue problem, Bulletin of the 
American Mathematical Society, Volume 56, 1950, pp. 115-139. 

2. Stone, M. H., Linear transformations in Hilbert space, American Mathematical Society, 

Colloquium Publication, New York, 1932, Ch. X, p. 484, Theorem 10.19. 

3. Friedrichs, K, 0., Uber die ausgezeichnete Randbedingung in der Spektraltheorie der halb- 

beschrdnkten gewohnlichen Differentialoperatoren zweiter Ordnung, Mathematische 
Annalen, Volume 112, 1935, pp. 1-23. On differential operators in Hilbert spaces, 
American Journal of Mathematics, Volume 61, 1939, pp. 523-544, and earlier papers 



K. O. FRIEDRICHS (S95) 449 

quoted there. Criteria for the discrete character of the spectra of ordinary differential 

operators, Courant Anniversary Volume, Interscience, New York, 1948, pp. 145-160. 
4. Titchmarsh, E. C, Eigenfunction Expansions Associated with Second-Order Differential 

Equations, Oxford, 1946, Ch. V, pp. 97-117. 
6. Kodaira, K., The eigenvalue problem for ordinary differential equations of the second order and 

Heisenberg's theory of S-matrices, American Journal of Mathematics, Volume 71, 

1949, pp. 921-945. 

6. Rellich, F., Die zuldssigen Randbedingungen bei den singular en Eigenwertproblemen der 

mathematischen Physik, Mathematische Zeitschrift, Volume 49, 1944, pp. 702-723. 

7. Hilbert, D., Grundziige einer allgemeinen Theorie der linearen Integralgleichungen, Nachrichten 

von der K. Gesellschaften der Wissenschaften zu Gottingen, vierte Mitteilung, 1906, 
pp. 157-227. See also the monograph of 1924, p. 147. 

8. Weyl, H., IJber beschrdnkte quadratische Formen, der en Differ enz vollstetig ist, Rendiconti del 

Circolo Matematico di Palermo, Adunanza del 13, December, 1908, pp. 1-20. ^ 

9. Sears, D. B., and E. C. Titchmarsh, Some eigenfunction formulae. The Quarterly Journal of 

Mathematics, Oxford, Second Series, Volume 1, 1950, pp. 165-175. 



Extension of WeyFs Integral for Harmonic 
Spherical Waves to Arbitrary Wave Shapes 



By H. PORITSKY 

General Engineering and Consulting Laboratory, 
General Electric Company 



1. Introduction 

In his paper of 1919 on the propagation of radio waves generated by a 
dipole near a flat earth, H. Weyl utihzed to advantage the representation of a 
sinusoidal spherical wave in terms of sinusoidal plane waves. With the point 




Figure 1 

source of the spherical wave at the origin, this representation for 2 > is given 
by the double integral 

(1.1) -ikR ^ 2^ J ^^P ' -ikoio^x + (3y -\- yz)] c?co, z > 0, 

where (see Figure 1) the integration is carried out over the half sphere 7 > 
of the unit sphere 

Paper presented at the June, 1950, Symposium on the Theory of Electromagnetic Waves, under 
the sponsorship of the Washington Square College of Arts and Science and the Institute for 
Mathematics and Mechanics of New York University and the Geophysical Research Directorate 
of the Air Force Cambridge Research Laboratories. 

33 (S97) 



34 (S98) 



ELECTROMAGNETIC WAVES 



(1.2) 



c^ + 0'+y' = h 



as well as over the complex portion of the unit sphere (1.2) corresponding to 6 
varying from to 7r/2 + ^ oo . This means that if a, /?, y, do) be expressed in 
spherical polar coordinates 6, (p: 



(1.3) 



a = cos (f sin d, 
(3 = sin (p sin 6, 
7 = cos d, 



dcx! = sin 6 dd d(p, 

then the (p limits in (1.1) are from to 27r, while the 6 limits are not from to 
Tr/2, but are given by 



(1.4) 



Dr* fl-/2 /» IT /2 + too n 

<J t/2 J 



as shown in Figure 2 in the complex ^-plane. These complex directions naturally 
do not show up in Figure 1. 



e PLANE 



r PLANE 

( r- COS e) 



Figure 2 



The corresponding values of y vary from 1 to 0, then to —i^, as sho^^'n on 
Figure 2. Since the path of integration of an analytic function may be distorted 



H. PORITSKY 



(S99) 35 



in the complex plane, the rectangular path may be changed into a path such 
as shown on Figure 2 by the broken line, from to 7r/2 -\r ico. 

By reflecting and refracting at 2 = each plane wave component of a 
similar representation in 2 < /i of a point source placed at (0, 0, h), Weyl ob- 
tained a similar double integral representation for the field of the dipole both 
above and in the ground. 

In evaluating the integrals in question, Weyl makes very effective use of 
changes of variables along the sphere (1.2) in spite of the partly real, partly 
complex range of integration. Thus, in proving (1.1) for any point P: (x, y, z), 
he introduces spherical coordinates R = {x^ -\- y^ -\- z^y^^, rj, yp with the line 
from the origin to the point P in question as polar axis, replaces the right-hand 
member of (1.1) by 



(1.5) 



i\y^L 



exp { —ikoR cos r?) sin ?7 drj, 



and introduces r = ikoR cos tj. The values of rj, cos t], t over the path of inte- 
gration are shown in Figure 3. The integral (1.5) becomes 



(1.6) 



i_ r 



dr = 



1 



—ikoR 



]ikoR -ikoR 
= ■^. . 
+ 00 XK/Qit 



The shifting of variables to 77, -^ is perhaps not as puzzling as the new 
choice of limits. This corresponds to the freedom of moving paths of inte- 



7 

PLANE 



COS^ 
PLANE 






I 
I 
I 



r 

PLANE 



il»oR 



Figure 3 

gration of a single variable in the complex plane; here the two-dimensional area 
of integration is distorted on the complex sphere (1.2), but without the benefit 
of a visual representation which is available for a single complex variable. 



36 (SlOO) ELECTROMAGNETIC WAVES 

However, since a spherical wave possesses spherical symmetry to start with, 
a shift of axes from that of Figure 1 to a similar set with OP as polar axis, is 
possible, and an equation similar to (1.1) may be set up for coordinate axes 
w^th OP as the 2;-axis, leading to (1.5) directly, rather than by transforming 
(1.1) from B, (p to ?;, \l/. 

In the follo^ving, we consider an extension of the Weyl integral given by 

(1 .7) m^M = -^ I F'[ct - {ax + &y + yz)] do,. 

The integral (1.7) represents a divergent spherical wave of arbitrary wave 
shape but with the inverse radius variation of amplitude, F{ct — R)/R, as a 
superposition of plane waves whose wave shape is F', the derivative of F, and 
of uniform amplitude per element of solid angle c?co presumably of the same 
half-sphere as in equation (1.1) and Figure 1, including both real and complex 
directions. 

Since F is defined originally as an arbitrary real function of its real argu- 
ment, whereas in (1.7) F' must range over complex values of its argument, the 
extension of F and F' to complex values must be first considered before (1.7) 
acquires a definite meaning. 

2. Derivation of (1.7) by Means of Fourier Integral Time 
Resolution, and by Assuming Analyticity of F 

One way of extending the Weyl integral to arbitrary spherical wave solutions 

(2.1) Fid - R)/R 

is by applying the Fourier integral in time. 
Recalling the exponential time factor 

(2.2) e'"\ ko = co/c, 
write (1.1) in the form 

^2 3) exp UU^t - R)] ^ _|2 I ^^p j .^^j^^ _ (^^ ^ ^^ ^ ^^y^ , ^,_ 

^^ext resolve F in F{ct — R) into a Fourier integral 

Aih) exp likoict - R)} dko . 
Utilizing (2.3) we obtain 
<2.5) F(ct - ^) = £ f -ikoAiko) dko J exp [iko[ct - (ax + /??/ + yz)]] da>. 



H. PORITSKY (SI 01) 37 

Now it will be noted that differentiation of (2.4) with respect to R yields 

rt+OO 

(2.6) F\ct - R) = ikoA{ko) exp {ikoict - R)} dko . 

J —CO 

Hence, carrying out the A;o-integration in (2.5) first, one presumably arrives at 
(1.7). 

In the derivation of (1.7) just given the Fourier integration over ko is 
carried out for real as well as complex values of argument [ct — {ax + /??/ + yz)], 
due to the complex portion of the sphere of integration, as explained in Figure 2 
and Section 1. Hence F', initially defined for real values, must be extended 
to complex values of its argument as an analytic function of its argument before 
equation (1.7) becomes unambiguous. This point will be examined in detail 
in Section 3. Moreover, it turns out that the integrals for negative k diverge. 

If by accident F can be extended to complex arguments ^ as an analytic 
function F(^) in the half-plane /(^) ^ 0, then a direct proof of equation (1.7) 
may be given, based on change of variables from 6, (p to rj, \p utihzed by Weyl 
in his proof of equation (1.1). With this change of variables, the right-hand 
member of (1.7) becomes 

1 /"IT /2 + t CD /»2t 

— — / F'{ct — R cos rj) sm 7} drj / dip 

Zir Jq Jo 

(2.7) 

= / F\ct — R cos TJ) d{cos ri) 

and introducing 

(2.8) T = -R cos 7] 



(2.9) i / "" F^ct + r)dT = 



Fjct - R) - F(^oo) 
R 



In order that the right-hand member of (2.9) reduce to the left-hand mem- 
ber of (1.7), it is necessary that 

(2.10) Fiioo) = 0, 
and it will be sufficient, if more generally 

(2.11) F(co) = 0. 

Indeed, since the derivative F' determines F only to within an additive constant, 
it is clear that equation (1,7) cannot possibly hold mthout some restriction on 
F, and the condition (2.11) will be used whenever possible. 

Upon closer examination, both proofs of (1.7) given above will be found 
insufficient. In the first place, the proof by means of the Fourier time resolution 
involves components with negative ko , and for such ko the exponentials factor 
exp {—ikoyz] in (1.1) leads to divergent integrals. Similarly, the direct proof 



38 (S102) ELECTROMAGNETIC WAVES 

based on the analyticity of F in the upper half-plane of its argument and the 
assumption (2.11), does not apply, in general, to cases when F is given as an 
arbitrary real function of its real argument. 

It is sho"\^Ti in Section 3 that while in general it is impossible to extend an 
arbitrary function F{Ji), defined for real values of ^, to complex values T = ^ + ^t?? 
as an analytic function of ^ over the whole ^-plane, it is always possible to break 
up F(Q into a sum of two functions 

(2.12) Fft) = hWm + F,({)], 

such that Fi(^) can be extended as an analytic function Fi(^) in ry > 0, while 
Fii^) cannot be so extended as an analytic function i^2(r) into 77 > but can 
be so extended in ?? < 0; moreover for real ^, F(f) is the real parfc of F-^{^) (as 
wellasof F2(rt). 

By applying the integral (1.7) not to F but to F^ , and taking the real part, 
one obtains an extension of the Weyl integral of general applicability, pro^-ided 
(2.10) or (2.11) applies to F^ . Actually, even this is not the case for the function 
F corresponding to a current pulse, and a further modification of the path of 
integration to handle these cases will be indicated in Section 4. 

3. Extension of Functions of a Real Variable to a Complex Variable 

The definition of the exponential integrand for complex values of the 
variables 7 or ^ in (1.1) offered no special difficulty. Similarly, no special diffi- 
culty is encountered in the interpretation of the refracted waves for the case 
where the angle of refraction, as given by SnelFs law, has a sine greater than 
unity, and similar remarks apply when, as shown in Figure 2, B is complex, 
since the definition of the exponential for all values of its argument, both real 
and complex, is clear enough. However, the interpretation of F' for complex 
values of its argument, when F is initially defined only for real values, is a real 
question that must be examined in detail. 

Since (1.7) was estabhshed formally by means of Fourier integrals, these 
integrals will be used to furnish the key to this extension of functions of a real 
variable to complex values of its variable. 

We consider, therefore, the extension by means of its Fourier integral of a 
real function ^(^), defined for real ^, to complex values of its argument obtained 
by replacing ^ by f = ^ + i-n- 

Write the Fourier integral of (^(Q in the form 

(3.1) <^(a = r\"'f(a)du. 

J —00 

Break up this integral into two parts 

(3.2) ^(9 = fe'-'Ka) da + f e"'f(a) da. 

Jo J-03 



H. PORITSKY (S103) 39 

It will be noted that if in the first integral ^ is replaced by ^, there results 

(3.3) [ e*"7W da= [ e'"^e-"7(«) da. 

For ?7 > this is convergent and represents an analytic function of f in the 
upper half ^-plane. A similar substitution in the second integral leads to a 
divergent integral. Only by replacing ^ by ^ = ^ — "^^^ will there result a 
convergent integral in 77 > 0. Hence we may extend (p{^) to the half-plane 
?7 > by means of 

(3.4) . <p= r e'^-'Ka) da + [" 6^^^%) da, v > 0, 

Jq J -co 

^ = k + i-ny ? = ^ - in- 

Thus defined, (p is not analytic in ^_in r? > 0, but is a sum of a function 
analytic in ^ and a function analytic in T- Furthermore (^ is a solution of the 
Laplace equation 

(3.5) S + & = 

for 7] > 0, and is thus the solution of the Dirichlet problem for the boundary 
value (p{^ for 77 > 0. The conditions for ip at infinity are treated by using 
circular inversion, say in the circle ^ -{- if' = 1, and transforming the infinite 
region into the neighborhood of the origin. 

An extension of (p(^) into the region ?; < by means of (3.4) is impossible 
since both integrals diverge in 77 < 0. In place of (3.4) one may use now 

(3.6) <p = f e^-'Ka) da + f e'^'fia) da. 

Jq J -03 

Remarks similar to those made about (3.4) apply to (3.6). However, the 
analytic function of f in 77 > in the right-hand member of (3.4) does not 
continue in 77 < into the analytic function of ^ in (3.6). Similarly, <p as defined 
by (3.6) is a solution of the Dirichlet problem for 77 ^ 0, taking on the values 
V? on 77 = 0, but (3.4), (3.6) do not constitute a solution of (3.5) in the whole 
f -plane and dip/drj is discontinuous on 77 = 0. 

In addition to the forms (3.4), (3.6) based on the Fourier integral (3.1), 
the function (p{^, rj) may also be represented by means of the integral 



(3.7) ^(?, ,) = ^ [ ° - 

TT J -co 77 



<p(i') d^' 



+ (« - r)- 



This is obtained by applying the Green's theorem to (p{^, 77) and the Green's 
function for either half plane. 



38 (S102) 



ELECTROMAGNETIC WAVES 



based on the analyticity of F in the upper half-plane of its argument and the 
assumption (2.11), does not apply, in general, to cases when F is given as an 
arbitrary real function of its real argument. 

It is sho^vn in Section 3 that while in general it is impossible to extend an 
arbitrary function F{^), defined for real values of ^, to complex values ^ = ^ + i'n, 
as an analytic function of f over the whole ^-plane, it is always possible to break 
up F(^) into a sum of two functions 



(2.12) 



F% = i[F,® + FM, 



such that Fi(^) can be extended as an analytic function Fi{^) in r; > 0, while 
F2U) cannot be so extended as an analytic function jPaCf) into ?? > but can 
be so extended in r? < 0; moreover for real f, F(^) is the real part of Fi(^) (as 

wellasof i^2(r)). 

By applying the integral (1.7) not to F but to F^ , and taking the real part, 
one obtains an extension of the Weyl integral of general applicability, pro^^ded 
(2.10) or (2.11) applies to F^ . Actually, even this is not the case for the function 
F corresponding to a current pulse, and a further modification of the path of 
integration to handle these cases will be indicated in Section 4. 



3. Extension of Functions of a Real Variable to a Complex Variable 

The definition of the exponential integrand for complex values of the 
variables 7 or ^ in (1.1) offered no special difficulty. Similarly, no special diffi- 
culty is encountered in the interpretation of the refracted waves for the case 
where the angle of refraction, as given by Snell's law, has a sine greater than 
unity, and similar remarks apply when, as shown in Figure 2, 6 is complex, 
since the definition of the exponential for all values of its argument, both real 
and complex, is clear enough. However, the interpretation of F' for complex 
values of its argument, when F is initially defined only for real values, is a real 
question that must be examined in detail. 

Since (1.7) was estabhshed formally by means of Fourier integrals, these 
integrals will be used to furnish the key to this extension of functions of a real 
variable to complex values of its variable. 

We consider, therefore, the extension by means of its Fourier integral of a 
real function (p(^), defined for real ^, to complex values of its argument obtained 
by replacing ^ by f = ^ + iv- 

Write the Fourier integral of ^(^) in the form 



(3.1) 



/ + 00 
e*'7(«) da. 
-00 



Break up this integral into two parts 

(3.2) <pi^) = f e"-'f{a) da + f e"'f{a) da. 



H. PORITSKY (S103) 39 

It will be noted that if in the first integral ^ is replaced by ^, there results 

(3.3) [ 6'"7(a) da= f e^-V-'Ka) da. 
Jo Jq 

For 7? > this is convergent and represents an analytic function of f in the 
upper half f-plane. A similar substitution in the second integral leads to a 
divergent integral. Only by replacing ^ by ^ = ^ — irj will there result a 
convergent integral in t; > 0. Hence we may extend (p{^) to the half-plane 
?; > by means of 

(3.4) . <p = r e'"'f(a) da + f e'"' J{a) da, v > 0, 

Thus defined, (p is not analytic in ^ in ?; > 0, but is a sum of a function 
analytic in ^ and a function analytic in f . Furthermore <p is a solution of the 
Laplace equation 

(3.5) ^ + ^. = 

for 77 > 0, and is thus the solution of the Dirichlet problem for the boundary 
value 9?(Q for rj > 0. The conditions for (p at infinity are treated by using 
circular inversion, say in the circle ^^ + ^^ = 1, and transforming the infinite 
region into the neighborhood of the origin. 

An extension of <p{^) into the region r; < by means of (3.4) is impossible 
since both integrals diverge in r; < 0. In place of (3.4) one may use now 

(3.6) <p = r e^-'Sia) da + f e'^'fia) da. 

Jq J —CO 

Remarks similar to those made about (3.4) apply to (3.6). However, the 
analytic function of ^ in r/ > in the right-hand member of (3.4) does not 
continue in 77 < into the analytic function of ^ in (3.6). Similarly, (p as defined 
by (3.6) is a solution of the Dirichlet problem for r; ^ 0, taking on the values 
(p on 7] = 0^ but (3.4), (3.6) do not constitute a solution of (3.5) in the whole 
f -plane and d<p/drj is discontinuous on r; = 0. 

In addition to the forms (3.4), (3.6) based on the Fourier integral (3.1), 
the function (p(^, rj) may also be represented by means of the integral 



(3.7) v(i, ')) = ^ r 






v+ii- n- 



This is obtained by applying the Green's theorem to (p{^, rj) and the Green's 
function for either half plane. 



40 (S104) ELECTROMAGNETIC WAVES 

In terms of the integrals (3.4) we now put 

I F,(f) = {" e-"7(«) da, r, a 
(3.8) 

For i; = 0, F,(9, F^i^) are conjugates of each other, equation (2.12) holdij. 
as well as the relations 

(3.9) F({) = (Re[F,(?)] = (Re[F,(^)]. 

In terms of the harmonic function tp defined by (3.7), we introduce F, by 
finding yp, the conjugate harmonic to </? in r; > and defining F^ as 

(3.10) F,(r) = ^ + iyp. 

4. Extension of the Weyl Integral 

For an arbitrary real function F we now put 

(4.1) F(|) = (Re[F,(|)] 

where F^ is analytic in i; > 0. We shall modify (1.7) as follows 

(4.2) F{c±^ ^ ^_^ I ^,j^^ _ ^^^ ^ ^^ ^ ^^^j ^J_ 

If the condition (2.10) applies to F, : 

(4.3) F,(i<«).= 

then the proof outHned in equations (2.7)-(2.9), applied to the right-hand mem- 
ber of equation (4.2), converts it to Fi{ct — R)/R and appHcation of equation 
(4.1) leads to equation (4.2). 

Of special interest is the case of a current pulse in the dipole, 7(0 = 5(0, 
since by DuhameFs theorem the field due to any current shape can be resolved 
into a superposition of the field due to current pulses. If the left-hand member 
of (4.2) represents the Hertz (vector) potential then F is proportional to the 
dipole moment (= /loo iit') dt') and is proportional to the Heaviside unit function 

(l for t > 0, 

(4.4) F{ct) = H{t) = < 

(O for ^ < 0. 

In this case the function F cannot be represented as a Fourier integral in view 
of convergence difficulties. The harmonic function (p{^, ri) is given by 

(4.5) <p{k, 77) = 1 - - tan-^ (V^) = 1 - (arg r)A 

TT 



H. PORITSKY 



(S105) 41 



and Fi (f ) is readily shown to be 



(4.6) 

while 
(4.7) 



F,(f) 



1 + - In f 

TT 



^!(f) = t,- 



Thus it will be seen that not only is equation (4.3) invalid, but /^i(rt even be- 
comes infinite at infinity. Actually, the condition (4.3) may be replaced by the 
less severe condition 



(4.8) 



(Sie[F,{io.)] = 0, 



but even this condition is not satisfied, the limit in question being 1/2. 

To overcome the difficulty just indicated, we modify the path of the 6- 
and r;-integrations from that shown in Figure 2 to the path shown in Figure 4 
and impose the condition 



(4.9) 



(Re[Fi(-oo)] = 0. 



It will be noted that this condition is satisfied by (4.6) as well as by Fi for cases 
where the dipole current vanishes for / less than fixed value to . 



PLANE 



( COS ^)- PLANE 



r - PLANE 



Figure 4 



Summarizing, it has been shown that a proper extension of (1.1) applicable 
to arbitrary spherical waves, and especially to waves started by a current pulse, 
is given by (4.2) with the path of integration for 6 and rj as shown in Figure 4. 



42 (S106) ELECTROMAGNETIC WAVES 

The integrand of (4.2) still has the advantage of being analytic in its arguments 
in the required domain of the latter. The interpretation in terms of plane waves 
is obvious, but essentially all the wave normals have complex direction cosines. 
It will be noted that only if the path of ^-integration is along the solid path 
of Figure 2 so that 7 is real or pure imaginary, will the argument 

(4.10) ct — (ax + /5?/ + yz) = ct — sin 6 {x cos (p -\- y sincp) — z cos 6 

of F[ in (4.2), when complex, lie in the upper-half plane. 

With the modified paths of Figure 4 and similar paths for 0, the quantity 
(4.10) no longer lies in the upper-half plane. In particular, if the ^-path is 
collapsed onto the pure imaginary axis, then cos B becomes real and > 1 and sin 6 
becomes positive imaginary, and (4.10) has a positive imaginary part only for 
certain combinations of x, y and z, but will acquire a negative imaginary part 
for other combinations of x, y, and z. 

One is thus faced with a choice of either using the restricted path of inte- 
gration shown as a solid line in Figure 2 and having to put up with an electro- 
static term 1/2R arising from infinity, or else removing the restriction of con- 
fining (4.10) to the upper half-plane. The latter can be done for the impulse 
current when F^ is given by (4.6), since F[{^) is analytic in the whole ^-plane 
except for ^ = 0. We shall adopt this second alternative, using the path of 
integration of Figure 4 for either 6 or 77, and allowing B or t] approach the pure 
imaginary axis. Care must be taken to avoid crossing T = 0- 

It is to be pointed out that the extension of the argument of F[{^) into 
^m (rt < is not to be confused with the function F'^i^) in that half -plane. 
Indeed, one readily shows that for the case when F[ is given by (4.7), F2 is the 
negative of F[ : 

(4.11) Fm = --,= -^i(r). 

The application of the above to a dipole near a flat earth will be presented 
in a future paper. 



Kirchhoff's Formula, Its Vector Analogue, and 
Other Field Equivalence Theorems 



By S. A. SCHELKUNOFF 
Bell Telephone Laboratories 



1. Introduction 



One of the purposes of this symposium is to discuss some of the theoretical 
difficulties involved in the solution of electromagnetic problems. For this reason 
I believe that the treatment of diffraction problems is an appropriate topic. 
In particular, the problem of radiation from a horn (acoustic or electromagnetic) 
has been treated approximately with the aid of one of several formulas sug- 
gested by Huygens' physical idea about wave propagation. These formulas are 
not equally powerful, do not always give equally good approximations, and do 
not inspire the same a priori confidence in the results. Of course, if no approxi- 
mations were made in the formulas, it would not matter which formula was 
used since then the result would always be exact. However, approximations 
are unavoidable except when the answer is already known and when there is 
no need for any formula. 

To understand how Huygens' physical assumption that the conditions at 
the front of a wave determine the subsequent wave motion suggests various 
formulas, let us consider a source first in an infinite homogeneous medium and 
then inside a perfectly rigid horn. In the first case the wavefronts are closed 
surfaces surrounding the source. In the second case, the wavefronts are open 
surfaces sliding along the walls of the horn until they reach the aperture. Event- 
ually, the wavefronts will become closed surfaces surrounding the horn as well 
as the source. Kirchhoff derived an explicit formula for the field outside a 
closed surface (S) in terms of the wave function and its normal derivative on 
(S) on the assumption that the source is inside (S) and that the medium outside 
{S) is homogeneous. In the case of the horn Kirchhoff 's surface of integration 
must enclose the horn as well as the source since the horn introduces a discon- 
tinuity in the medium. This closed surface may be chosen to consist of an 
"aperture surface" (So) together with the exterior surface of the horn. In the 
case of electromagnetic waves the present writer proved another theorem [1, 2] 
(the 'induction Theorem") in which the surface of integration is solely the 
aperture surface (Sa). An obvious analogue of this theorem for scalar waves 

Paper presented at the June, 1950, Symposium on the Theory of Electromagnetic Waves, under 
the sponsorship of the Washington Square College of Arts and Science and the Institute for 
Mathematics and Mechanics of New York University and the Geophysical Research Directorate 
of the Air Force Cambridge Research Laboratories. 

43 (S107) 



44 (S108) ELECTROMAGNETIC WAVES 

will be discussed in this paper. To understand the Induction Theorem we should 
note that the conditions at a wavefront before it reaches the aperture are the 
same whether the horn is truncated at the aperture or extended indefinitely. 
Hence, we might expect the existence of a formula which expresses the field of 
a truncated horn in terms of the field of an infinite horn. The Induction Theorem 
provides such a formula. The differences between Kirchhoff's formula and the 
Induction Formula are the following: {1) in the former the surface of integration 
must completely enclose the horn while in the latter the surface of integration 
is an open surface either in the interior or in the aperture but such that together 
with the boundary of the horn it forms a closed surface, (2) in Kirchhoff's 
formula the integrand depends on the actual field of the truncated horn while 
in the Induction Formula it depends on the field which exists when the horn 
is extended to infinity/ {3) in Kirchhoff's formula the integrand depends also 
on the Green's function for an infinite homogeneous medium while in the In« 
duction Formula the Green's function is for the medium with the horn (that is, 
the Green's function must satisfy the proper boundary conditions at the surface 
of the horn). At first it may seem that the latter condition restricts the useful- 
ness of the Induction Formula. This, however, is not the case. In the Induction 
Formula we are given a distribution of virtual sources over the aperture which 
in the presence of the horn produces the correct external field of the truncated 
horn and the field returned into the horn by the truncation. The calculation 
of the field of the virtual sources may be divided into two separate problems. 
First we can calculate the field on the assumption that the sources are in an 
infinite homogeneous medium; then we impress this field on the horn and try 
to evaluate the reflection from the horn. In the case of large apertures we shall 
find that the field impressed on the horn is relatively small except near the edge. 
There are other formulas which express the effect of an alteration in the 
environment of a source (such as truncation of a horn) on the field, or the 
equivalence of fields produced by two different systems of sources. For example, 
if an electric source is inside a perfectly conducting closed surface, the field 
outside is zero. Hence, the currents in the surface produce an external field 
which is equal and opposite to that of the source. Consequentl}^, the currents 
equal and opposite to those in the surface produce the same external field as 
the original source. This is a field analogue of Norton's circuit equivalence 
theorem, in which a network with interior generators is replaced by a source 
of fixed current in parallel with the admittance of the network. Similarly there 
is a field analogue of Helmholtz-Thevenin's circuit equivalence theorem in which 
a network with interior generators is replaced by a source of fixed voltage in 
series with the impedance of the network. In the field analogue certain hypo- 
thetical magnetic currents correspond to sources of fixed voltage. There are 
also circuit analogues of Kirchhoff's formula and of the Induction Formula. 



^The actual field of the truncated horn would also give the correct result, but it would 
serve no useful purpose to employ it. 



S. A. SCHELKUNOFF (S109) 45 

These various formulas are interesting from the theoretical point of view 
and useful from the practical point of view. Kirchhoff's formula is the oldest 
of all. It has been used successfully to obtain first approximations to the solu- 
tions of problems involving transmission of sound and light through large 
apertures in screens. However, in more recent years it has been found that 
attempts to apply Kirchhoff's formula to the calculation of electromagnetic 
radiation from horns led to various difficulties. For instance, Barrow and 
Greene [3] found that either the polarization or the intensity of the radiation 
field involved substantial errors when the formula was applied to one or another 
field vector. Of course, some polarization errors are also involved in optical 
applications; but these errors are appreciable only in directions making large 
angles with the principal beam, and in these directions optical fields are ex- 
tremely weak. In microwave radio, on the other hand, the apertures are not 
particularly large compared with the wavelength, and large angles have to be 
considered. 

There are two possible sources of error. Kirchhoff's formula requires inte- 
gration of certain functions over a closed surface (S); but in applications we 
are forced to integrate over an ' 'aperture surface" (So) which is only a small 
part of the total surface (S). It has been suggested [4] that having obtained 
the first approximation as usual we could use this approximation in Kirchhoff's 
formula to obtain a second approximation. But in the next section we shall 
presently see that the second approximation obtained in this way must equal 
the first and that the initial error remains intact. To resolve this difficulty we 
need the Induction Formula. 

In the electromagnetic case, the second source of error might be in that 
Kirchhoff's formula involves scalar wave functions and not vector wave func- 
tions. It is true that each cartesian component of either E or H is a scalar 
wave function, and thus Kirchhoff's formula may be applied to it, but the 
various components are not independent wave functions since they must also 
satisfy Maxwell's equations. Thus when we apply Kirchhoff's formula to a 
single component and integrate over only a part of the total closed surface we 
may (and actually do) obtain a solution of the wave equation which does not 
satisfy Maxwell's equations. To overcome this difficulty, Love [5] obtained a 
vector analogue of Kirchhoff's formula which expresses an electromagnetic field 
at a given point in terms of the tangential components of E and ^ on a closed 
surface (S). Later another derivation of the same formula was given by 
MacDonald [6]. However, if Love's formula is applied to only a part of the 
closed surface {S), the result still does not satisfy Maxwell's equations. The 
present writer obtained [1] a simpler form of the vector analogue of Kirchhoff's 
formula. For a closed surface this form is mathematically equivalent to Love's 
form; but when our formula is applied to a portion of the closed surface, the 
result satisfies Maxwell's equations. Stratton and Chu resolved the same diffi- 
culty in another way [7]. When applying Love's formula to a portion of the 
closed surface, they add certain line integrals to the original surface integrals. 



46 (SI 10) 



ELECTROMAGNETIC WAVES 



These line integrals are suggested by the physical interpretation of the integrand 
in Love's formula and do not follow from the mathematical proof. 

However, none of these vector analogues of Kirchhoff's formula can be used 
in the manner suggested by Born [4] to obtain a higher order approximation. 
The initial error made in the integrand of either Kirchhoff's formula or its vector 
analogue must remain intact throughout repeated applications of the formula. 
To resolve this difficulty we need the Induction Theorem. 

There are other difficulties inherent in Kirchhoff's formula (and its vector 
analogue) which are best explained by referring to its explicit analytical formu- 
lation. 

2. Kirchhoff^s Formula 

There are two forms of Kirchhoff's formula, one vahd only for a finite 
region and the other applicable to either a finite or an infinite region. First let 
us consider a finite region. Let uhe Si solution of the scalar wave equation free 



1 




Fig. 1. A closed surface (S) and reference point P inside (S). 



from singularities anywhere inside a closed surface (*S), Figure 1. The value 
of u at any interior point P is determined by its value and the value of its normal 
derivative on (S). The best known formula is 



(1) uiP) 



'IK' 



d exp {—il3r] 



dn 



47rr 



du exp 
dn 



i^r] 



47rr 



)dS, 



= 27r/X. 



This formula is usually obtained from Green's theorem [4, 8, 9]. But this proof 
also gives [10] 

(2) n(P) = jf ( 

(-S) 

and more generally 



d cos jSr _ du cos_§r\ , ^^ 



dn 47rr 



dn 47rr 



■) 



(3) 
where 

(4) 



'(^) = //(^^-|^)'^^ 



cos jSr , .sin /3r 



47rr 



47rr 



J 



S. A. SCHELKUNOFF 



(Sill) 47 



and A is an arbitrary constant, li A = i, we have (1) but with advanced 
potentials instead of retarded potentials. What this ambiguity means is that 
at any interior point P 



(5) 



//(» 



d sin /3r 
dn 47rr 



dusm 
dn 



47rr 



dS = 0. 



(S) 



The constant A does not affect the result, but, we must stress, only if we use 
the exact values of u and its normal derivative on (S). To apply these formulas 
to the problem of radiation from a horn we can choose {S) as shown in Figure 2, 





Fig. 2. A closed surface, Sa + Sc , surrounding Fig. 3. A closed surface surrounding 
reference point P outside a horn. a horn with internal sources. 

where (S) consists of an ''aperture surface" (Sa) and a complementary surface 
(Sc) so that (So) + (Sc) is a closed surface surrounding the region of interest. 
We can also choose {S) = (Sa) + (Sc) in such a way that it surrounds 
the horn completely. Figure 3. If we assume that the sources are inside the 
horn, we know more about their field than we did in the preceding case. Now 
we know not only that uisa. solution of the wave equation, free from singularities 




Fig. 4. Two closed surfaces, Sa + Sc and Sa, , surrounding an acoustic horn. 

outside (S) but also that u behaves at infinity as r~^ exp { —i(3r}. We are not 
permitted to apply Green's formula directly to the infinite region outside (S); 
but we can apply it to the finite region between {S) and another surface (So.) 



48 (SI 12) ELECTROMAGNETIC WAVES 

which surrounds (*S), Figure 4. As we try to remove (S^) to infinity we find 
that the contribution from this surface does not vanish unless the constant A 
in equation (2) equals —i. In this case we obtain equation (1). In the pre- 
ceding case we could not make use of the behavior of u at infinity for the simple 
reason that the region of interest did not extend to infinity. 

It is now clear that we should be careful how we apply Kirchhoff's formula 
to the problem of radiation from an acoustic horn. Sometimes the argument 
proceeds as follows: Choose a closed surface (S) consisting of two parts, one 
(So) in the aperture and the other (*SJ so as to enclose the point P where we 
wish to determine the excess pressure u. The exact values of u and du/dn are 
not known on (S); but if the aperture is large, then physical intuition tells us 
that the wave motion in the aperture is likely to be about the same, except 
possibly near the edge, as in a hypothetical case in which the horn is continued 
indefinitely. The problem of an infinite horn can be solved; hence it is assumed 
that we know fairly accurately u and du/dn on the aperture surface (So). Some- 
times it is also assumed that on (^SJ both u and du/dn vanish. With these 
assumptions the following first approximation to u{P) is obtained from (1) 

(-S„) 

where Uq and du^/dn are the values corresponding to the infinite horn. If we 
apply exactly the same arguments to equation (2), we find another first ap- 
proximation 

(Sa) 

For any number of different values of A in (4) we can obtain corresponding 
approximations from (3). These various approximations are far from being 
nearly the same; they are radically different. Experience shows that for large 
horns equation (6) gives very good results. Equation (7), on the other hand, 
gives totally wrong results. Unless we are willing to accept (6) as a semi- 
empirical formula, we have to revise the arguments which led to (6) and (7) 
as being equally permissible approximations. We can make use of the phj^sical 
principle enunciated by Huygens and decide in favor of (6) on physical grounds; 
but this can hardly be satisfactory unless we can establish a connection between 
the physical principle and equation (1). The derivation of (1) from Green's 
formula does not establish such a connection. 

Going over the argument leading to the approximation (6), we observe that 
it is not really necessary to assume that ii and du/dn vanish over {&/)■ Our 
practical success with this approximation means that while these values ob- 
viously do not vanish, their integrals over {S/) must be small. Somehow we 
should be able to prove it. We can do this by choosing {S) as in Figure 3. Here 
our physical intuition tells us that the excess presure u on (*S<.) is small except 



S. A. SCHELKUNOFF (SI 13) 49 

near the edge. From the boundary conditions we know that the normal velocity, 
and therefore du/dn, vanishes on {Sc). All this makes it plausible on physical 
grounds that (6) should be a satisfactory approximation. This is as far as we 
can go if we base our argument on Kirchhoff's theorem. Presently we shall see 
that the Induction Theorem enables us to arrive at (6) on purely mathematical 
grounds. 

Next let us see if we can improve on the first approximation — at least in 
principle — by using Kirchhoff's formula (1) repeatedly. We choose a closed 
surface {S') very near {S) in the region to which (1) is applicable. From (6) 
we calculate Ux and dui/dn on {S'). Let us substitute these values in (1) and 
calculate 2/2 (-P). Is U2{P) a better approximation to u{P) than Ui{P)? The 
answer is no for the simple reason that U2(P) = Ui{P). Once we have calculated 
Ui{P) from (6), we have a solution of the wave equation which is free from 
singularities in the region under consideration; by Kirchhoff's formula this 
solution is recovered from its values and the values of its normal derivative on 
(S'). This argument can be presented in a different form. When we approxi- 
mate equation (1) by (6) we in effect replace a system of sources (1) which is 
truly equivalent to the given source inside the horn by another system of sources 
(2) which is only approximately equivalent. Thereafter, if we apply Kirchhoff's 
formula to the field produced by system (2), we must recover the same field. 
In fact, if we could obtain increasingly better approximations by using Kirch- 
hoff's formula repeatedly, we should find ourselves in an embarrassing position 
of being able to lift ourselves by our own boots. There is nothing in the formula 
that contains a self-correcting mechanism. 

3. Field Equivalence Theorems 

Field equivalence theorems express the identity, in certain specified regions, 
of the fields produced by given sources and the fields produced by appropriate 
sources on the boundaries of the regions. Equation (1) is within the meaning 
of this definition; but equation (2) is not. Since we are interested in physical 
applications, we shall state the theorems in physical terms; but it is not difficult 
to rephrase them in strictly mathematical terms. A point source will become 
a singularity of a specified kind. A continuous distribution of sources on a 
surface will become a specified discontinuity either in the wave function or in 
its normal derivative or both. The field ''produced" by such a discontinuity 
is the field which has this discontinuity and which behaves at infinity as 
exp { — ^■/3r j /r. 

In the case of acoustic waves we have two types of elementary sources: 
(1) a pair of pistons moving in opposite directions, Figure 5a, and (2) a single 
piston oscillating back and forth. Figure 5b. Across the double piston the 
pressure Uq is supposed to be continuous and the particle velocity |o = ~" (V^^Po) 
duo/dn discontinuous by the amount |o.n ~~ lo^.n ; hence, the double piston acts 
as a source of volume current (^o.n ~ ^o.n) dS, where dS is the area of one face 



50 (SI 14) 



ELECTROMAGNETIC WAVES 



of the piston. Across the single piston the particle velocity is continuous but 
the pressure is discontinuous by the amount {Un — u~) dS. In the case of 
electromagnetic waves the elementary sources are: (1) a surface element of 



.^zz. 



fo'nOS 




77777777777. 



ZjIA. 



r 



////,' //y.^/ //'//<' 



77777 




Fig. 5. Two elementary sources of acoustic waves: (a) a pair of pistons mo\'ing in opposite 
directions, (b) a single oscillating piston. 

electric current of Hnear density C« , Figure 6a, and (2) a surface element of 
magnetic current of hnear density C^ , Figure 6b. Across the electric current 





Fig. 6. Two elementary sources of electromagnetic waves : (a) a surface 
element of electric current, (b) a surface element of magnetic current. 

element the tangential electric intensity is continuous and the tangential mag- 
netic intensity discontinuous. The moment of the electric current element is 
Ce d& — nifll — HZ) dS, where n is the normal to the element. Across the 
magnetic current element the tangential magnetic intensity is continuous and 
the tangential electric intensity is discontinuous. The moment of the magnetic 
current element is 

C^ dS = (^o" - Eo)n ds. 
We can now state the following theorems. 



4. Acoustic Field Equivalence Theorems 

Theorem 1. The Acoustic Induction Theorem 

Consider a rigid or a perfectly pliable horn of finite dimensions with a given 
system of internal sources, Figure 7a, producing a field of excess pressure u. 



S. A. SCHELKUNOFF (SI 15) 51 

Let (Sa) be an ''aperture surface" which together with the boundary (S^) of 
the horn forms a closed surface. Let Uq be the excess pressure produced by the 
same sources when the horn is continued indefinitely, Figure 7b. The field 

/' 
/* 

/ 

/ 

'\ 

*\ 

Fig. 7. A horn with internal sources: (a) horn is finite and closed through an aperture 
surface (>Sa), (b) horn is extended indefinitely. 

'produced in the presence of the truncated horn by a layer of elementary sources of 
volume current |o.n dS = — (1/^copo) duo/dn and a layer of elementary sources of 
excess pressure Uq dS, also on (So), equals the field u produced by the given sources 
at points external to (Sa) -\- (Sh). At points internal to (So) + (Sh) the field of 
the aperture layers equals the difference u — Uq between the field produced by the given 
sources in the finite horn and the field produced by the same sources in the extended 
horn. 

If G is the excess pressure produced by a source of unit volume current in the 
presence of the given horn, then the above theorem states that 

(8) u(P) = jj Gio^poiLn dS) + jj ^uo dS 

iSa) (.Sa) 

iSa) (.Sa) 

if P is an exterior point, and 

(9) uiP)=u.iP)- jJG'-^dS+ jj'£u.dS 

(Sa) (Sa) 

if P is an interior point. 

In particular, if the boundary of the horn is conical, both before and after 
its extension to infinity, and if the junction between the duct and the horn is 



52 (SI 16) ELECTROMAGNETIC WAVES 

small compared ^^dth the wavelength, or if an acoustic lens is introduced in the 
junction to transform the plane equiphase surfaces in the duct into spherical 
in the horn, then Uq = A exp { —i^ro}/ro , where Tq is the distance from the apex 
of the horn. Let us choose (So) to be a segment of the sphere of length / centered 
at this apex and passing through the edge of the aperture, then, 

(10) u{P) = -A Jf G f^^^^^^f^ dS + A fj^i^^^T^dS. 

(So) (5„) 

Note the following differences between the Induction Theorem (8) and the 
Kirchhoff formula (1) as far as practical applications are concerned: (1) in (1) 
we know the Green's function G = exp { — ^/?r ) /47rr but we do not know the 
distribution of sources on the surface of integration, while in (8) we know the 
distribution of sources but not the Green's function; (2) in (1) the surface of 
integration is a closed surface surrounding the given sources while in (8) it is 
an open surface in the aperture. In a later section we shall discuss the practical 
consequences of this difference. 

To prove the induction theorem we start "wdth the extended horn, Figure 
7b. Let Uq be the excess pressure produced by the given sources. Let u be the 
excess pressure produced by the same sources in the truncated horn, Figure 7a. 
We shall now consider the excess pressure given by 

u = u — Uq , in region (1), 

(11) 

= u, in region (2) . 

Since u and Uq satisfy the wave equations, u also satisfies the wave equation 
everywhere except on the boundary (Sa) between the regions (1) and (2). At 
infinity u varies as exp {—i^r}/r because u varies in this manner. In region 
(1) u has no singularities because the sources of u and Uq are supposed to be the 
same. On the surface of the horn u satisfies the required boundary conditions 
because u and Uq satisfy them; across the boundary (Sa) u and dtl/dn are dis- 
continuous 

(12) u" - u~ = Uo , (du^/dn) - {du~/dn) = (duo/^n). 

These discontinuities define the sources of u as stated in the theorem. Since a 
given system of sources produces a unique field, we are certain to recover u 
defined by (1) from the known discontinuities (12). 

We can also prove this theorem as follows. Starting wdth the field Uo in 
the extended horn, we assume that the aperture surface (Sa) is a '^perfect ab- 
sorber" defined in such a way that it does not affect the field in region (1) and 
reduces it to zero in region (2). Across (Sa) the new field is discontinuous. The 
increments in the excess pressure and its derivative are respectively —Uo{S) 
and — dUo{S)/dn. These discontinuities define the sources on (So) which together 
^\'ith the given sources produce Uq in region (1) and the zero field in region (2). 



S. A. SCHELKUNOFF (SI 17) 53 

The extended part of the horn may now be removed since it is in a field-free 
region. If we superimpose on the present field the field produced by the dis- 
continuities Uo{S) and dUo{S)/dn on (Sa), we obtain the field which has no dis- 
continuities on (Sa). Thus we obtain the field for the truncated horn in the 
form given by equations (8) and (9) . 

While Figure 7b illustrates the "analytic" continuation of the horn and a 
simple type of aperture surface, the proof of the theorem holds for any con- 
tinuation of the horn and any aperture surface as long as this surface joins the 
boundary of the horn. We require only that jointly with the boundary of the 
horn the aperture surface should enclose the source, Figure 8. In addition to an 
arbitrary continuation of the boundary of the horn we may introduce obstacles 
in region (2). Nevertheless, if we determine Uq and duo/dn and from these 
values obtain the equivalent layer of sources over (Sa), then this layer wdll 
produce the correct field in region (2) and the difference between the correct 
field and Uq in region (1) even after we alter region (2) in any way whatsoever. 
We may remove obstacles altogether or in part or merely alter their shape. We 
may remove the obstacles and insert other obstacles. We may alter the medium 
itself. But we may not make any alterations in region (1). 

Theorem 2. Kirchhoff's Equivalence Theorem 

Consider a closed surface (S), such as (So) + (Sc) in Figure 2 or in Figure 
4, which separates the medium into two regions, one containing the sources and 
the horn and the other source-free. Let u be the excess pressure created by the 
given sources. In the source-free region this field equals that produced by a layer 
of elementary sources of volume current ^n dS = — {l/icopo) (du/dn) on (S) and a 
layer of elementary sources of pressure vdS, also on (S). In the remaining region 
the field produced by these sources vanishes. On account of the second part of 
this theorem the horn can be removed without disturbing the field of the virtua 
sources on (S). Therefore in computing this field we may use the Green's 
function for the infinite homogeneous medium; that is, we obtain equation (1) 
for points in the source-free region. 

Proofs are analogous to those for Theorem 1. We consider the wave func- 
tion u which vanishes identically in the region originally containing the given 
sources and the horn, and which equals u in the source-free region. Then we 
show this field satisfies all the requirements of the field produced by the virtual 
sources on (S) as defined in the theorem. Alternatively, we can postulate a 
perfectly absorbing layer coinciding with (S), determine the distribution of sinks 
on it, and then eliminate the sinks by superimposing an equal distribution of 
sources but with opposite signs. Thus we recover the original field. 

We note that in the above proofs it makes no difference whether (S) is of 
the type shown in Figure 2 or in Figure 4, while the proof based on Green's 
formula requires separate consideration of the two cases. The present proofs 
are more elastic because they are independent of particular analytic expressions 
and can be carried through under more general conditions. On the other hand, 



54 (SI 18) ELECTROMAGNETIC WAVES 

by the above methods we cannot derive equations (2) or (3) as easily as from 
Green's formula. 

It may seem remarkable that the field produced by the virtual sources 
defined in the above theorem vanishes in the region which originally contained 
the horn and the given sources. A prescribed distribution of sources on a surface 
must produce a field which has given discontinuities in u and du/dn across the 
surface; but normally it is impossible to tell beforehand what values ^x\\\ be 
assumed by u and du/dn on the two sides of the surface. But under the condi- 
tions of the present theorem the sources on {S) are not arbitrary; they are ob- 
tained from the values of u and du/dn belonging to a self-consistent field possess- 
ing special properties. 

Theorem 3 

Consider a horn with given interior sources and an aperture surface (*SJ, 
Figure 7a or Figure 8, which together with the boundary of the horn forms a 
closed surface. Let Uq be the excess pressure for the case in which {S^) is abso- 
lutely rigid. Then, the excess pressure produced by the same sources in an open 
horn equals the sum of Uq and the excess pressure produced by a layer of elementary 

i 



0) yv \j 

\ 

Fig. 8. An extended horn with irregular boundary and an obstacle. 

sources of pressure Uq dS distributed over (So). If, as in Theorem 1, G is the excess 
pressure produced by a source of unit volume current, then the excess pressure 
produced by the given source in the open horn is 

(13) u = Uo+ jl ^^^odS. 

(Sa) 

Note that each of the terms on the right is discontinuous across (aSJ, but their 
sum is continuous. The derivative of the first term is zero on (So) and that of 
the second term is continuous. 

The proof of this theorem is analogous to the proofs of the preceding theo- 
rems. We start with the field given by Uo , determine the discontinuities in Uo 
and duo/dn across {S^), and remove them by superimposing the field with equal 
and opposite discontinuities. 



S. A. SCHELKUNOFF (SI 19) 55 

Theorem 4 

This theorem is the dual of the preceding theorem. // Uq is the excess pressw e 
for the case in which (So) is absolutely pliable, then 



(14) u = Uo - jj G 



dn 



5. Electromagnetic Field Equivalence Theorems 

Electromagnetic field equivalence theorems are similar to the above. Thus 
we have 

Theorem 1. The Electromagnetic Induction Theorem 

Consider a perfectly conducting^ horn of finite dimensions with a given 
system of internal sources, Figure 7a, producing an electromagnetic field E, H. 
Let (So) be an ''aperture surface" which together with the boundary (Sh) of 
the horn forms a closed surface. Let Eq , Hq be the field produced by the same 
sources on the assumption that the horn is extended indefinitely, Figure 7b. 
The field produced in the presence of the truncated horn by an electric current sheet 
of density n x Hq and a magnetic current sheet of density Eq X n, both on (S^), 
equals E, H at points external to (So) + (Sh) and E — Eq , H — Hq at the internal 
points. 

To prove, let 

E = E - Eq , H = H - Hq , in region (1), 



(15) 



= £', = H, in region (2) . 



Since Ej H and Eq , Hq are solutions of Maxwell's equations, E, H is also a 
solution. At infinity E, H vary as exp [—i^r]/r because E, H vary in this 
manner. In region (1) E, H has no singularities because the sources of E, H 
and Eq , Hq are supposed to be the same. On the surface of the horn E, H satisfy 
the proper boundary conditions because E, H and Eq , Hq satisfy these conditions. 
Across the boundary {S^ between the regions (1) and (2), the tangential com- 
ponents of E, H are discontinuous by the amount defined in (15); thus 

(lo) -C^tan -ti^tan = -t^'O.tan j -"tan -^tan = -f^O.tan • 

According to Maxwell's equations these discontinuities imply magnetic and 
electric currents on {S^ whose linear densities are respectively 

(17) Cm = EqXu, C, =nX Hq . 



2In theory the boundary could be either a perfect electric conductor defined by JE'taa = 
or a perfect magnetic conductor defined by i^tan = 0. 



56 (S120) ELECTROMAGNETIC WAVES 

Since the field produced by a given system of sources in the presence of given 
boundaries is unique, we are able to recover E, H from (17) as stated in the 
theorem. 

The proof is seen to be analogous to the corresponding proof of the scalar 
induction theorem. Similarly the second proof of the scalar theorem can be 
carried over to the vector case. 

Theorem 2. The Field Equivalence Theorem for Free Space 

Consider a closed surface (S), such as (Sa) + (Sc) in Figure 2 or Figure 3, 
which separates the medium into two regions, one containing the sources and 
the horn and the other source-free. Let E, H be the field created by the given 
sources. In the source-free region this field equals that produced hy an electric 
current sheet of density n X H and a magnetic current sheet of density E x n, both 
on (S). In the remaining region the field produced hy these sources vanishes. 

The proof is so similar to the proof already given for the acoustic case and 
for the electromagnetic induction theorem that we need not repeat it. 

The analytic expression for the fields produced by given electric and mag- 
netic currents may be found elsewhere [11]. We cite the results for convenience, 



E = —ic-fjiA + (l/zcoe) grad div A — curl F, 
H = curl A + (l/icciJi) grad div F — iue F, 



(18) 
where 



(5) 



This is the form in which we expressed the vector analogue of Kirchhoff's 
formula (1) in our earher papers [1, 2]. An alternative form given by Love [o], 
MacDonald [6], Stratton and Chu [7] is 



(20) 



E = -jj [iccixin X H)xly + (nXE) XV4^ + {n-E)\7i] dS, 

(S) 

H = jj licc€(n X E)4^ - (nX H) XVrP - (n'H)ViP] dS, 



where 

(21) 4' = e-'^y47rr. 

This formula is obtained from the vector analogue of Green's formula and in 
the derivation for a closed surface of the type shown in Figure 1, ^J/ could equally 
well be given by equation (4). To obtain the particular form of equation (20) 
in which xp is given by (21) from Green's formula we must — as in the correspond- 



S. A. SCHELKUNOFF (S121) 57 

ing scalar case — consider two closed surfaces which enclose the horn as in Figure 
4; then we can prove that for any field produced by sources inside (So) + (S^) 
the contribution from (So,) vanishes, provided yp is given by (21). 

All our remarks concerning Kirchhoff's scalar formula apply equally well 
to its vector analogue. There is one added difficulty in connection with equa- 
tion (20). If (S) is replaced by the aperture surface (Sa), then E and H do not 
satisfy Maxwell's equations. To remedy this Stratton and Chu add certain line 
integrals. This solves one difficulty but raises another. What is the reason 
for adding these line integrals, aside from wishing to obtain expressions which 
satisfy Maxwell's equations? If we add arbitrarily these line integrals in the 
case of vector fields, why not add them also in the scalar case? From the phy- 
sical point of view the line integrals are understandable, since they represent 
electric and magnetic charges that may be associated with the surface currents. 
But in the derivation of (20) from Green's formula, E and H in the integrand 
represent the surface values of the field intensities. More than Green's formula 
is needed to identify n x H and E X n with the equivalent sources on {S). 
Thus, in practice, we really need the stronger form of the theorem given by 
equations (18) and (19). 

Theorem 3 

Consider a perfectly conducting horn with given interior sources and an 
aperture surface (Sa), Figure 7a, which together with the boundary of the horn 
forms a closed surface. Let Eq , Hq be the field produced by these sources when 
(Sa) is a perfect electric conductor. Then, the field produced by the same sources 
in an open horn is the sum of Eq , Hq and the field produced by an electric current 
sheet of density n X Hq over (Sa). 

This is the field analogue of Norton's circuit equivalence theorem. 

The proof is analogous to that for the corresponding acoustic case (Theorem 
4). The field Eq , Ho is zero outside^ (Sa) + (Sh). Since (Sa) is a perfect con- 
ductor, £^tan vanishes on it and thus is continuous across (Sa). On the other 
hand, Ht^^^ is discontinuous and the amount of the discontinuity gives the 
current density —(nX Hq) on (Sa). If we superimpose on Eq , ^o the field 
produced by the electric current sheet of density (n X Hq) on (Sa) we ehminate 
the discontinuity in ^tan on (Sa) and maintain the continuity of E'taa • The 
resultant field satisfies all the boundary conditions for the field produced by the 
given sources in the open horn. 

Theorem 4 

This is the dual of Theorem 3. 

If Eq , Hq is the field produced by the given sources when (Sa) is a perfect mag- 
netic conductor, then the field produced by the same sources in the open horn is the 



^It is to insure this property that we have to assume that the boundary of the horn is 
either a perfect electric conductor or a perfect magnetic conductor. There should be no leakage 
through the boundary. 



58 (S122) ELECTROMAGNETIC WAVES 

sum of Eq , Hq and the field produced by a magnetic current sheet of density E X n 
on (SJ. 

This is the field analogue of the Helmholtz-Thevenin circuit theorem. There 
is also a circuit analogue of our Induction Theorem 1 but it is neither well kno^n 
nor particularly useful. In the field case, on the other hand, the analogues of 
the Norton and Heknholtz-Thevenin theorems are more interesting than useful 
while the Equivalence Theorem and the Induction Theorem acquire real im- 
portance. 

6. Possible Methods of Obtaining Higher Order Approximations to 
Radiation Fields of Horns 

For most practical purposes the first approximation for the radiation 
fields is satisfactory [12, 13, 14]. But if we are interested in the second approxi- 
mation, we may be able to obtain it from the Induction Theorem. According 
to this theorem we have to evaluate the field of kno^n ''aperture currents" in 
the presence of the horn. We can consider this field as the sum of the free 
space field of the same currents and the field refiected from the horn. The free 
space field is given exactly by (19) and (20) where (S) = (Sa). If we evaluate 
this field at the surface of the horn, particularly the tangential component of 
E, we shall be able to estimate the strength of the reflected field. In the case 
of large apertures the primary field of the aperture currents is throTMi forward 
and the reflected field must be relatively small except near the edge of the aper- 




FiG. 9. A circular horn, illustrating the horn region (h), the complementary horn 
region (ch) and the free space region (Js). 

ture. The primary field induces electric currents in the walls of the horn. In 
terms of the unknown density of these currents we can express their field, using 
(19) and (20). Equating the tangential component of E for this field to the 
negative of the tangential component of E for the primary field, we obtain one or 
two integral equations for the reflected field. Integral equations of this kind are 
not very tractable; but we would be trying to evaluate only a small correction 
term to the major part of the field given by (19) and (20). 



S. A. SCHELKUNOFF (S123) 59 

In the case of circular horns, Figure 9, it is more practical to obtain the 
field of the aperture currents by successive approximations using expansions in 
spherical harmonics. As shown in the figure we can subdivide the entire space 
into three regions: the horn region (h), the free space region (fs), and the com- 
plementary horn region (ch). Assuming at first that the aperture currents are 
in free space, we obtain the field in all three regions. We now take the tangential 
field on the boundary between (fs) and (ch) and expand it into spherical har- 
monics appropriate to the region (ch). Similarly, we take the tangential field 
on the inner surface of the boundary between (fs) and (h) and expand it into 
harmonics appropriate to the region (h). Thus we obtain new fields for (h) and 
(ch) which satisfy proper boundary conditions on the surface of the horn. We 
then consider these fields as fields impressed on the free space region. In this 
way we obtain the second approximation to the radiation field of the horn. 



Bibliography 

1. Schelkunoff, S. A., Some equivalence theorems of electromagnetics and their application to 

radiation problems, Bell System Technical Journal, Volume 15, 1936, pp. 92-112. 

2. Schelkunoff, S. A., On diffraction and radiation of electromagnetic waves, Physical Review, 

Volume 56, 1939, pp. 308-316. 

3. Barrow, W. L., and Greene, F. M., Rectangular hollow-pipe radiators, Institute of Radio 

Engineers, Proceedings, Volume 26, 1938, pp. 1498-1519. 

4. Born, M., Optik, Julius Springer, Berlin, 1933, p. 152. 

5. Love, A. E. H., The integration of the equations of propagation of electric waves. Philosophical 

Transactions, Volume 197, Series A, 1901, pp. 1-45. 

6. MacDonald, H. M., The integration of the equations of propagation of electric waves, London 

Mathematical Society, Proceedings, Volume 10, 1911, pp. 91-95. 

7. Stratton, J. A., and Chu, L. J., Diffraction theory of electromagnetic waves, Physical Review, 

Volume 56, 1939, pp. 99-107. 

8. Bateman, H., Partial differential equations of mathematical physics. University Press, Cam- 

bridge, 1932, p. 189; Dover Publications, New York, ]944. 

9. Baker, B. B., and Copson, E. T., The mathematical theory of Huygens^ principle, Oxford 

Press, 1939, p. 25. 

10. Webster, A. G., Partial differential equations of mathematical physics, G. E. Stechert, New 

York, 1927, p. 221. 

11. Schelkunoff, S. A., Electromagnetic waves, D. Van Nostrand, New York, 1943, pp. 128-132, 

eq. (1-8), (1-10), and (2-13). 

12. Watson, R. B., and Horton, C. W., The radiation patterns of dielectric rods — experiment and 

theory, Journal of Applied Physics, Volume 19, 1948, pp. 661-670. 

13. Watson, R. B., and Horton, C. W., On the calculation of radiation patterns of dielectric rods. 

Journal of Applied Physics, Volume 19, 1948, pp. 836-837. 

14. Horton, C. W., On the theory of the radiation patterns of electromagnetic horns of moderate 

flare angles. Institute of Radio Engineers, Proceedings, Volume 37, 1949, pp. 744-749. 



I 



On the Diffraction Theory of Gaussian Optics 



By H. BREMMER 
Philips Research Laboratories, N. V. Philips' Gloeilampenfabrieken, 
Eindhoven, Netherlands 



1. Introduction 

The diffraction theory of optical systems is usually developed with reference 
to objects having a periodical structure or to an object consisting of a single 
point. On the other hand, when investigating the resolving power of the system 
it is necessary to introduce at least two separate point objects. For other ques- 
tions, such as the connection of Abbe's intermediate image with the Fourier 
transform of the structure of the object, it is even necessary to consider arbitrary 
objects. The importance of considering arbitrary object structures is also evident 
from papers by Gabor [1] and Toraldo di Francia [2] concerning electron micros- 
copy and phase microscopy respectively. 

In this paper, too, the diffraction theory of optical imaging will be de- 
veloped for objects with arbitrary structure. However, the familiar theory will 
here be based on rigorous solutions of the wave equation instead of the con- 
ventional approximation of Kirchhoff's formula. Apart from mathematical 
elegance, such a treatment is of advantage when such questions as the distri- 
bution of the wave function in the neighbourhood of the object and of its 
paraxial image plane are dealt with. In the first solution to be treated the 
similarity of the wave functions in the object plane and in the corresponding 
paraxial image plane (Gaussian systems with unlimited aperture) proves to be 
connected with Neumann's integral theorem for Bessel functions instead of the 
Fourier identity as in the Kirchhoff approximations. While assuming an illumi- 
nation by a plane wave arriving along the optical axis of symmetry we shall 
discuss in succession the wave function in the object space and in the image 
space. It will be shown how another solution may account for the effects of 
optical aberrations and of limited apertures. 

2. The Transmission Function of the Object 

The illuminating primary plane wave arriving along the optical axis of 
symmetry (;2-axis) will be given as exp {iikoZ — o)t)}. We assume the wave 
number A^o with a positive (possibly infinitely small) imaginary part. The 

Paper presented at the June, 1950, Symposium on the Theory of Electromagnetic Waves, under 
the sponsorship of the Washington Square College of Arts and Science and the Institute for 
Mathematics and Mechanics of New York University and the Geophysical Research Directorate 
of the Air Force Cambridge Research Laboratories. 

61 (S125) 



62 (S126) 



ELECTROMAGNETIC WAVES 



complex value of ko is in accordance ^^ith an attenuation of the primary^ wave 
in the direction from z = — c»to0= +co (direction of propagation); in what 
follows the time factor exp { —io^t] ^^11 be omitted throughout. 

The introduction of a single wave function does not prevent the apphcation 
to vectorial problems. For such problems the wave function under discussion 
may represent any quantity satisfying the ordinary wave equation {A -{- kl)u = 
in the space outside the imaging system. Thus our wave function can refer to 
any of the field components of E and H in the case of electromagnetic waves. 
The final differing beha^dour of the various components then follows from a 
suitable modification of the object and of the space inside the imaging system. 
These modifications depend, amongst other things, on the mutual connections 
between these components according to ^Maxwell's equations. For electron 
optics u should be identified ^\ith Schrodinger's \^-function. 




Figure 1 



As usual we assume a two-dimensional infinitely great non reflective object 
at right angle to the axis of sj^mmetry. The plane of the object will be taken 
as the plane z = (see Figure 1). The primary wave exp {ikoz} is then propa- 
gated undisturbedly in the space z < only and shows the constant value 1 
when arri\dng at the object. The modification of the wave function by the 
object can be described by gi^ang the distribution of the wave function directly 
behind the object (z = 0+), viz. 



(1) 



u(x, ij, + ) = 1 + e(.r, y). 
is a complex- valued function.^ Its de^'iation from zero ac- 



In general 

counts for the attenuation as well as for the phase retardation caused bj^ the 
object. For purely absorbing objects e is real (and negative). In vectorial 
problems e will be different for the various field components; thus it is also 
possible to include polarization effects due to the object. Our description is 
also apphcable to objects of finite thickness if the plane of reference z = is 
taken again directly behind the object. 

^The function e corresponds to Gabor's function t{x, y); see loc. cit., p. 459. 



H. BREMMER (S127) 63 

3. The Wave Function in the Object Space 

The most important part of this space extends between the object plane 
and the front of the imaging system. In its region we can completely ignore 
the imaging system if the latter is supposed to be non reflective. The determi- 
nation of the wave function u in the object space then amounts to solving the 
wave equation (A -{- kl)u = under the following boundary conditions: 

(a) u is equal to the given transmission function 1 + e(Q) = u{Q) at each 
point Q of the object plane, 

(b) u vanishes at infinity, if 2 > 0, proportional to exp likR]/R (R = distance 
from the field point to the origin of the coordinate system). 

This problem can be solved with Green's method which leads to the formula^ 

(2) «(^) = -2^i//^0..(Q)^SLi^, 

where (IOq is a surface element of the object plane and QP the distance from 
the object point Q to the point under consideration P. The boundary condition 
(1) holds according to the general identity 

® ^ i . "- i: II ''O" ^'(® sLiMi = ^(p), ,„ ,„ > 



ep-*zQ 



which has to be applied here with the upper signs when Zq = 0. 

Formula (3) expresses the sifting property of the following two-dimensional 
impulse function: 

(4) ^ ^ lini - ; 2 'j_ J ^72^— ^ = Kx)8(y) = -f^ , ^mk>0 

in which p = (x^ + y^^^^ while (p^ + z^^^ is defined positive. The vahdity 
of the limit (4) (which vanishes everywhere beyond p = 0) is easily proved by 
an integration over p from to <» . 

Identity (3) is of great importance in our diffraction theory and can also 
be considered as a geometrical interpretation of Neumann's integral theorem^ 

(5) F(P) =^l duu jj dOo F{Q) Jo{u-PQ), 

in which P is supposed to be situated in the Q-plane (a surface element of the 
latter is denoted by dOg). The equivalence of (3) and (5) is demonstrated by 
substituting Sommerf eld's integral for exp {iko-QP}/QP in (3). 

Returning to the representation (2) for the wave function in the object 
space we observe that this formula may be interpreted as a superposition of 

^Compare C. J. Bouwkamp, A contribution to the theory of acoustic radiation, Philips Re- 
search Report 1, 1946, p. 251. 

^See G. N. Watson, A Treatise of Bessel Functions, Cambridge, 1944, p. 475; compare 
Bouwkamp, loc. cit., p. 23. 



64 (S128) ELECTROMAGNETIC WAVES 

the field of dipoles situated in the object plane, the surface density of these 
dipoles being proportional to the transmission function u{Q), Finally we empha- 
size the ver}^ general significance of (2). Indeed, this formula is appHcable to 
the determination of the wave function in any half space containing no sources, 
if the distribution of this function has been given over the plane boundary' of 
the half space. In other words, (2) constitutes the mathematical formulation of 
Huygens' principle for a non-curved surface. 

4. The Wave Function in the Image Space in Front of the Paraxial 
Image Plane for Gaussian Systems with Unlimited Aperture 

The wave function in the image space (the space beyond the imaging 
sj'stem) depends, amongst other things, on the properties of the imaging system. 
These properties are always given in t-erms of geometrical optics and therefore 
enable us to construct a geometric-optical approximation. In the case of a 
Gaussian system this approximation proves to be a rigorous solution of the 
wave equation and can be taken as the final solution in the case of an unlimited 
aperture. 

In the above theory we considered a two-dimensional object. We can 
imagine geometric-optical rays diverging from each object point Q. these rays 
being transformed into convergent beams by the imaging system. We have to 
consider the rays leaving a single object point first. Moreover we confine our- 
selves provisionally to an infinitely small pencil of the rays radiating from such 
an object point Q. The cross-section of such a pencil changes along its tra- 
jectory, the variation being connected ^Wth that of the wave function. This 
follows from the law of conservation of energy. Indeed, in isotropic media 
the vector of the energy current density is directed along the tangents of the 
trajectory so that the energy radiated at Q into the pencil can never escape 
from it. On the other hand, the amplitude of the energy current density is 
proportional to the square of the modulus | u \ of the wave functions satisfjdng 
the scalar wave equation. The conservation of the energ\^ current through the 
pencil under consideration is thus expressed by the condition 

(6) \u\'^ da = constant 

along the entire trajectory. 

The above considerations apply to any optical systems; now we concentrate 
on Gaussian systems. Here a point source at Q in the object space corresponds 
in the image space to a point source at the paraxial image point Q' oi Q. Special 
normalization thus leads to the combination: 

^^^^ exp {iko^QP} ^ p ^ ^^.^^^ gp^^^^ 

/^i N A exp l—iko'Q'P] T^ • • ^ 

(/b) A — /p — -, P m image space; Zp < Zq^ . 



H. BREMMER 



(S129) 65 



The different signs in the exponents are in accordance with the propagation 
away from Q in the object space and towards Q' in the first half of the image 
space. The value of | A | follows at once from a comparison of the situation at 
the two points of intersection Pi and P2 of the above pencil with the two principal 
planes of the imaging system (see Figure 2). The Gaussian properties of the 




Figure 2 

latter involve dap^ = dap, (in the usual approximations of the paraxial theory, 
which neglects the inclination of the pencil with respect to the axis of sym- 
metry). Next we infer from (6) that | u{Pi) \ = | u^Pi) \, or in virtue of (7) 

I A I = Q'P,/QP, . 

This ratio represents the paraxial magnification A^ (again neglecting the 
inclination of the rays) so that | A | = A^. It is reasonable to assume (7b) with 
a phase factor in accordance with the concept of an optical path length in- 
creasing continuously along the ray-trajectory QP. This phase factor proves 
to be exp {iko-QP}, QP being defined as the optical distance: 



Kq J Q 



k(s) ds, 



while k is different from kp inside the imaging system only. The proportionality 
of ( 7b) w ith exp {iko-QP} involves a factor exp {iko{QP + PQOl = ©xp 
{iko Q PQ^} w hich is to be contained in A. Evidently, the total optical path 
length QPQ' is independent of P, all the rays connecting Q and Q' having the 
sa me o ptical length (Fermat's principle) so that QPQ^ may simply be marked 
as QQ\ 

Independent of the phase changes connected with the optical distance, the 
wave function may change its sign in some point (in the imaging system) which 
corresponds to a zero amplitude of this function. An additional phase factor 
exp {iir} = —1 then has to be added. This happens for instance for the z- 
component of the E vector in the case of a converging optical lens. In fact, 
the transversality and continuity of E (see Figure 2) imply opposite signs of 
E^ , apart from the changes due to the phase factor exp {iko-QP}, at either 
side of the optical system. 

Remembering tha t \ A \ = N, we thus arrive at two possible values of A, 
viz. =bA^ exp {iko-QQ'}. The solution (7) may accordingly be replaced by 



66 (S130) ELECTROMAGNETIC WAVES 

exp liko'QP 
(8) 



^p , P in object space; Zp > 0, 



P in image space; Zp < Zq' . 

We return to formula (2) for the complete wave function in the object 
space. Leaving out of consideration the operator d/dzp , the amplitude of 
the contributions due to the point source at Q is given by 

-{1/2t) dOc,u{Q). 

According to the above considerations this corresponds to a contribution of 

for the wave function in the image space Zp < Zq> . A final integration over 
the object plane while also taking into account the operator d/dZp yields the 
following expressions for the wave function in the first part of the image space 
(for an object at z = illuminated by the plane wave exp {ikoz}): 



(9b) 



Zp < Zq: 



With respect to this formula we would make the following remarks: 

(a) (9) represents a rigorous solution of the wave equation, being the super- 
position of a number of dipoles situated at the various points Q\ Therefore 
we may consider (9) to be the exact expression for a Gaussian system ^ith 
unlimited aperture; 

(b) the parameter N has been put before the integration sign. This is allowed 
because the paraxial theory involves a magnification that is independent of the 
special situation of the point Q in the object plane; 

(c) the sign of (8) is taken to be the same for all object points Q, so that this 
very sign could be put in front of (9). This involves that the wave function 
under consideration should have a zero point either on any trajectory con- 
necting the finite object with P or on none of these trajectories. This condition 
is satisfied for the field components usually considered. 

5. The Wave Function in and beyond the Paraxial Image Plane 
(Unlimited Aperture) 

The wave function in these parts of the image space has to be a continuation 
of the expressions (9) . Formula (9b) is particularly suited for the determination 



H. BREMMER (S131) 67 

of u in the paraxial image plane z = Zq' itself when the integration is extended 
over this plane instead of over the object plane ^ = 0. This change of the domain 
of integration is effected by the transformation 

^Q - ^ J 2/0 - ^ 

which indicates that the distances between the points Q' are A^" times larger 
than those between the corresponding points Q (see also Figure 1). At the 
same time we obtain the new surface element 

(IOq' = dxQ' dijQ' = N^ dxq dyg = N^ dOq . 

In this way (9b) is transformed into 

(10) u(P) = ^^^JI dO,. X u{Q) exp [ih-W] X ^xp {-»fco-Q'P| ^ 

Zp -^ Zq' . 

Now let P approach some point Pq of the image plane z = Zq' . The value 
of u{Pq) then follows from formula (3) in which we should apply the lower 
signs because P is approaching the plane of integration from the left. More- 
over A;o has to be replaced by —ko in (3), which is permissible for real k^ . Evi- 
dently the function F to be substituted in (3) is given by 

1 



F(Po) = =F ^^(Po) exp {^/co•P;Po!, 

Pq indicating the object point having Pq as paraxial image. The final result 
of the limiting procedure becomes 

(11) u{Po) = =F ^ ^(^o) exp [ih-P^o], zp^ = zq, . 

The similarity of the distributions of the wave function in the object plane 
and in the conjugated image plane here results from Neumann's integral theorem 
for Bessel functions which is essentially identical with formula (3). 

As for the space beyond the paraxial image plane, (9a) proves to be still 
valid i f its sign is inverted. This can be verified as follows. For Zp > Zq> , 
QP = QQ' + Q'P holds as will be clear from the fact that, on its way, the ray 
connecting Q and P has to pass necessarily along Q\ The transformation of 
(12a) into an integral over the plane z = Zq^ , then leads to the following ex- 
pression instead of (10): 

exp {iko-Q'P} 



u{P) = ±^£- // dO,> X u{Q) exp {ih-QQ'] x ^^^^^ 



> Zc 



Point P may now approach the plane z = Zq. from the back. This implies 
an application of (3) \\4th the upper signs which leads once more to (11). Sum- 
marizing we have the following formulae in the image space mth integrations 
extending over the object plane: 



68 (S132) 



ELECTROMAGNETIC WAVES 



(12) u{P) 



li-JjdO.uiQ)'-^^''-^^^ 



Q'P 



^-^uinexp {iko'P'P}, 



Zp < Zq. 



Zp — Zq' 



Zp > Zq. 



It is remarkable that the continuous wave function in the image space has to 
be represented by two different formulae at either side of the image plane. For 
a single object point (corresponding to u{Q) = 5(xg — Xo)8{yQ — yo)) the 
different sign can be interpreted as a phase-shift tt accompamdng the passing 
along the image point, i.e. along the focal point of the beam in the image space. 
The attention to phase shifts of this type has been dra^\Ti by Debye. 

6. Derivation of a Rigorous Solution of the Wave Equation 
for Gaussian Systems with Limited Aperture 

The remark made at the end of Section 3 ^\'ith respect to Huygens' prin- 
ciple applies to the image space as well as to the object space because the wave 
equation is valid in either of them. Thus we can also compute a wave function 
in the image space beyond some plane z = Zk by substituting in (2) the wave 
function distribution u{K) instead of u{Q), K being an arbitrary- point oi z = 
Zk . This procedure enables us to account for the effect of an aperture in a plane 
perpendicular to the symmetry-axis. While identifying this plane with z = Zk , 
we take iiiQ) equal to the undisturbed wave function inside the aperture, and 
equal to zero beyond it. Therefore Ave can substitute (9b) for u{K) inside the 
boundarv' of the aperture, because the latter is always situated between the 
optical system and the paraxial image plane. 

In this way we get the four-dimensional integral 

d exp \iko-KP} 



"^^) = -£^JJ 



dOK 



(13) 



jj dOou(Q) exp likoQQ' 



KP 



d exp l-iJcn'Q'K 



Zk < Zp 



<Z, 



dzK QK 

dOx being a surface element of the iv-plane, the integration of which has to be 
stopped at the boundary of the aperture. 

Contrarily to (12), the solution (13) is represented by a single formula for 
the entire image space beyond the plane z = Zk . Either solution satisfies the 
rigorous wave equation and accounts for the geometric-optical properties of 
the Gaussian system. 

7. Modification of the Wave Function in Some Intermediate Plane 

The derivation of (13) is particularly important because it can be extended 
so as to include also the effects of artificial modifications imposed on the diffrac- 



H. BREMMER (S133) 6^ 

tion pattern in some plane perpendicular to the axis. This intermediate plane 
may coincide with the above aperture plane z = Zk which we still assume to 
be situated between the imaging system and the image plane. The above 
modifications can be described by a function (p{K) the modulus and phase of 
which correspond to the attenuation and phase retardation effected in the K- 
plane. The function (p{K) may be compared with the transmission function of 
the object (see Section 2). Evidently (p{K) has to be added as a factor to the 
i^-integrand of (13) in order to account for the modifications under considera- 
tion. Moreover, we are free to invert the order of integrations in this integral; 
thus we get the two following identical extensions of (13): 



u{P) = ±£, jj dO^^iK) 
(14a) 



K ff ^n .z^^AexpJi^oi^ 
dzp KP 



• jj dOQ u{Q) exp {tko'QQ ] — — ^7^ 

uiP) = ± £2 // dOo u{Q) exp {iko'Q^} jj dOK <p{K) 

(14b) 

d expliko-KP] d exp {-iko'Q'K} 
dzp KP dzK Q'K 



Zk ^ Zp ] Zk '^ Z 



Q' 



The physical interpretation of the inversion of the order of integration in 
similar expressions has been emphasized by Zernike [3]. In our case the inner 
integral of (14a) represents the diffraction pattern established in the i^-plane 
by the complete object. Therefore we can interpret (14a) as the imaging of this 
diffraction pattern. In (14b), however, the imaging of a single object point 
constitutes the starting point, the final integration referring to the superposition 
of the effects of the individual object points. 

In practice, for instance in phase microscopy, the focal plane in the image 
space is often used as intermediate plane. The inner integral of (14a) or (13) 
then represents the well-known intermediate image of Abbe if the artificial 
modifications are absent {<p{K) = 1 inside the aperture). Further, the effect 
of the aperture itself is included in (14) by considering the integration as ex- 
tending over the complete i^-plane and taking (p{K) equal to 1 inside the aperture 
and equal to zero beyond it. Thus a circular aperture with radius a and centre 
on the optical axis is described by 

^{K) = U{a' -x\- yl), 

U being the unit function {U(x) = 1 for a: > and U\x) = for a; < 0). Even 
the theory of optical aberrations can be included by the introduction of a similar 
function (p, which, however, in that case in general depends on both K and Q 
(compare Section 9). 

The practical significance of (14b) may be illustrated by the example of a 
small absorbing phase plate as used in phase microscopy. If we assume this 



70 (S134) ELECTROMAGNETIC WAVES 

plate to be infinitely small and situated at the point Kc^ on the axis of symmetn% 
the modification is given by 

<p{K) = 1 + ^A5fe)5(^/^). 

Substitution of this in (14b) yields the ordinary undisturbed wave (corre- 
sponding to (p = 1) and a secondary wave given by 

^ lAN _d_ exp liko'KoP} ff ,^ 
^ 4x^ dz^ KoP JJ "^^^ 

(15) 

• u{Q) exp [iko-QQ ] ^ ^/^^ ^ • 

This wave can be interpreted as arising from a dipole at Kq . If i^o co- 
incides ^^'ith the focal point F2 in the image space, we can, moreover, apply 
the relation 

(16) W =0F, + F,Q', 

which follows from a consideration of the trajectory QF2Q' and in which OF 2 is 
the optical distance along the axis between object plane and focal plane. Evalua- 
tion of (15) vrith. the aid of (16) leads to the follo^^ing simplified expression for 
the secondary wave generated by the phase plate in the focal point: 

T ^ ko-F,0 -exp {tio-OF,! — L_ jj ^0, u{Q) ^t^tt— • 

^Tien approximating F2Q' by F2O' we find the well-kno^Ti proportionaHty 
of this wave ^vith the average transmission of the object. 

8. Transition to the Usual Approximations 

Once more we assume the intermediate or aperture plane to be identical 
T\-ith the focal plane {z = Zp) . The usual approximation for the wave function 
in the image space is then obtained from (14b) by appljdng the foUoTsing sub- 
stitutions based on the assumptions that the distances KP and KQ' are large 
compared to the wavelength and that the inclinations of KP and KQ' with 
respect to the axis of symmetry are small: 

(a) ?7:o for d/d Zp and d/d Zr ', 

(b) Zp — Zp and Zq. — Zp for KP and KQ' respectively in the denominators; 

(c) KP^Zp-Zp+ 2{zp-zp) 
and the corresponding expression for KQ'; 

(d) e«^-^ + lr^' 



H. BREMMER (S135) 71 

00' being the optical distance along the axis between object plane and image 
plane, and / the focal length. 

Moreover we apply the follo\\'ing relations of Gaussian optics: 

Xq, = Nxq ; Vq' = NyQ ; Zq> - Zp = -Nf. 

The final result, 



Nkl exp likloo' + .p - z^, + #4:^1 1 re rr 

-(^) = ^ U. -..)(.. -J" ^"^^^ // '^^ ^^«) // '^^ 



•(p{K^) exp < — ^/bo 



(? + .-;^K + fr + i7^K 






exp <[^ [(xp - iVa:o)xx + (2/p - A^2/o)2/ir]|. 



2 W/ 

then represents approximately the wave function in the image space (for a 
Gaussian system) corresponding to an object at 2 = and an illuminating 
beam given by exp {ikoz}. The function (p{K) describes modifications imposed 
in the focal plane of the image. 

This general formula can be simplified considerably for P in the paraxial 
image plane (zp = Zq>), the simplification being based on the cancelling of the 
quadratic terms in the exponent in the integrand. In this case the final ex- 
pression becomes 

u(P) = T ^° 'Xiy ^' // dOo u{Q) ff dO, v{K) 
(17) 

For (p{K) = 1 (unlimited aperture) formula (17) reduces to a Fourier integral 
resulting in expression (11) which was derived above from the solution (10). 
Because we started in this section from solution (14) or (13), the equivalence 
of either solution is confirmed here in the case of an unlimited aperture as far 
as the usual approximations are concerned. 

9. Extension to Non-Gaussian Systems 

For these systems too we can derive a geometric-optical expression for the 
wave function in the image space by applying the arguments of section 4. 
Contrarily to Gaussian systems, however, this expression constitutes no solution 
of the w^ave equation. Nevertheless it can be considered the saddle-point ap- 
proximation of such a solution as will be discussed now. 

We again start from the point-source solution exp {iko-QP}/QP in the 
object space. The modulus | w | of the corresponding geometric-optical approxi- 
mation in the image space has to be derived from formula (6) which concerned 



72 (S136) ELECTROMAGNETIC WAVES 

the conservation of the energy current inside a narrow pencil of trajectories. 
We call dup a surface element at P of the wave front passing through this point. 
This surface element constitutes also the cross-section of an infinitely small 
pencil of raj^s leaving Q. The spatial angle c/12, of these rays is constant through- 
out the trajectory across the image space as follows from the rectilinear course 
of the rays. Considering the properties of the Gaussian curvature K(P) of the 
wa^-efront at P we derive 



d(Tp = 



K{P) 



Calling c?fi^ the constant value of the spatial angle of the pencil under 
consideration in the object space we deduce from (6) by comparing the values 
oi \ u\^ da in the object space and at P: 

Hence : 



u{P) I = 



^«(^)li 



if Q,PJ 



the index Q indicates reference to a special object point. 

We choose the phase of u in accordance with the optical distance QP from 
Q to P. Thus we arrive at the following geometric-optical approximation 
which replaces formulae (7) and (8) for Gaussian systems: 

(18a) exp [ik^QP] ^ p ^ ^^.^^^ ^p^^^ 

(18b) ±[^o(^)(fi") ^J''' e^P {iko'QP}, P in imagespace. 

The expression (18b) proves to be the saddle-point approximation of the 
integral 

(19) J = d= ^ exp {iko'H{Q)} jj m. dQ,Y'' exp {iko-PPw] 



W(Q) 



the integration of which extends over an arbitrary wave front in the image 
space for the trajectories leaving Q; PPw represents the distance from P to the 
tangent plane of this wave front in any of its points TF (see Figure 3), H{Q) 
the optical distance from Q to an arbitrary point W. The verification of (18b) 
as the saddle-point approximation of (19) is facihtated by the introduction of 
a rectangular co-ordinate system ^-q^ with its origin at the point of intersection 
Po of the ray QP with the wave front. We take the ^-axis along PqP, and the 
^-axis and rj-axis along the tangents of the lines of curvature passing through 
Po . As to the curvature properties we can replace the wave front near Po by 
an enveloping quadric which has the following equation: 



r 



2 \p, ^ pj' 



H. BREMMER 



(S137) 73 



Pi and p2 being the radii of curvature at Pq . In the ^rj^ system the distance 
PPw reads: 

pp _ r>j:> _ "-^0 ' Pi >-2 _ PPq ~r P2 2 

Zpi Zp2 

from which we deduce the saddle-point vahies ^w = Vw = for the exponential 
factor in (19). In other words, the saddle point coincides with Pq . The in- 
tegral (19) now reduces to (18b) if the expression 

Id^KdUi]''' WTT.X r^^.T'' dap 



d^ drj 
is replaced by its saddle-point value 



= KiW) ^ 






df dri 



^^^"lUflXj - p,p, LidflXj 



at any point W. The verification still depends on the geometrical relation 

1 



K,{P) = 



{PoP + pOiPoP + P2) ' 



which holds according to the generation of the wavefront through P by a shift 
of the points W over a distance PqP along the normals (see Figure 3) . 




Figure 3 



The integral (19) is composed of plane waves with wave number ko and 
thus determines a rigorous solution of the wave equation. Therefore we replace 
(18) by the following corresponding solutions in object space and image space: 



(20) 



exp {iko'QP 

QP 



27r 



^ exp \iko-H{Q)] jj m^ dQ.y' exp [iko-PPw], 



W{Q) 



the saddlepoint approximations of which satisfy the necessary geometric-optical 
conditions. 

Returning to our problem of an object at 2 = illuminated by the plane 



74 (S138) ELECTROMAGNETIC WAVES 

wave exp {ikoz}, we still have to apply the operator — (l/2Tr){d/dZp) Jf dOg u(Q) 
• • • , in accordance wdth (2). Thus we get the following rigorous solution in 
the image space for the diffraction field due to an object illuminated by a plane 
wave arriving along the axis: 

u(P) = ::p'^^-l-jj dO, u{Q) exp {iko'H{Q)\ 
(21a) 

. jj [dn, dn^Y'' exp {iko-PPw} . 

W(.Q) 

Besides, we have the less accurate formula based on (18b) instead of (20): 
(21b) uiP) -^ii-JjdO, „(Q)[x«(P)(g)^ J'" exp {ik.-m. 

As in section 7 we consider an intermediate plane z = Zr . The distribution 
of the wave function existing in that plane according to (21) may be written 
formally as 

/'oo^ fTr\ ^^ ^ ff ^n rn\ fn ^r^ exp {ih(M - KQ')] 

(22) u{K) = ^ ^^jj (^^Q '^(QMQ, K) —^ — ^^g^T ^-^ 

if (p{Q, K) has been defined; e.g. when applying (21b) 

.(Q,z)==w[i^.(i^)rx^[(g)^J^ 

(23) _ _ 

X exp {ih{QK - QQ' + KQ')]. 

According to formula (2) we arrive again at expressions (14) for the field 
beyond the intermediate plane if (p{K) has been replaced by (p(Q, K). This 
function, however, must always be put in the inner integral o^^ing to its de- 
pendence on both Q and K. Of course (p{Q, K) should reduce to unity for a 
Gaussian system mth unlimited aperture. In the case of the geometric-optical 
approximations this is evident from the three factors indicated in (23). These 
factors account for the non-Gaussian deviations connected in succession with: 

(a) the curvature of the wavefronts in the image space, 

(b) the ratio of the spatial angles of one and the same pencil in object space and 
image space. 

(c) the optical distance from Q to the point K in the intermediate plane. 

Bibliography 

1. Gabor, D., Microscopy by reconstructed wave-fronts, Proceedings of the Royal Society of 

London, Series A, Volume 197, 1949, p. 454. 

2. Toraldo di Francia, G., II fenemeno di Gibbs nella microscopia in contrasto difa-se, II Xuovo 

Cionento, Volume 6, 1949, p. 30. 

3. Zernike, F., Phase contrast, a new method for the microscopic observation of transparent 

objects, Physica, Volume 9, 1942, p. 686. 



Diffraction and Reflection of Pulses by Wedges 

and Corners^ 



By JOSEPH B. KELLER and ALBERT BLANK 

Mathematics Research Group, 

Washington Square College, 

New York University 

1. Introduction 

The diffraction and reflection by a perfectly conducting wedge of a periodic 
plane wave (with wavefront parallel to the edge) has been treated by MacDonald 
[1], who obtained a series of Bessel functions for the solution. We will consider 
the corresponding problem for an incident plane pulse. This problem could 
be solved by employing a Fourier Integral of MacDonald's solution. However, 
the solution can be obtained directly as an explicit closed expression in terms 
of elementary functions. This is possible because the solution for this geometry 
is ''conical" and independent of ''radial" distance in xyt-spsice, and this allows 
separation in appropriate coordinates. (This is Busemann's conical flow method 
[2] widely used in supersonic aerodynamics.) The method can also be used in 
problems for which the periodic solution is not known. 

Before the conical flow method can be employed, it is necessary to know 
how the plane discontinuity surface propagates. The propagation of such 
discontinuities has been investigated by R. K. Luneberg [3] in electromagnetic 
theorj^, and by J. B. Keller [4] in acoustics. It is found in both cases that the 
discontinuity surface satisfies a first order partial differential equation, the 
eiconal equation in homogeneous media, and that the magnitude of the dis- 
continuity varies in a simple manner as the surface moves. We make use of 
these results in the present investigation, and they enable us to convert the 
initial-boundary value problem into a characteristic-boundary value problem in 
xyt-spsice. The conical flow method is then used to obtain the solution. 

The results apply to a single component of electric or magnetic field parallel 
to the edge of a perfectly conducting wedge or corner {v = on wall corresponds 

Paper presented at the June, 1950, Symposium on the Theory of Electromagnetic Waves, under 
the sponsorship of the Washington Square College of Arts and Science and the Institute for 
Mathematics and Mechanics of New York University and the Geophysical Research Directorate 
of the Air Force Cambridge Research Laboratories. 

*This work was performed at Washington Square College of Arts and Science, New York 
University, and was supported in part by Contract No. AF-19(122)-42 with the U. S. Air 
Force through sponsorship of Geophysical Research Directorate, Air Force Cambridge Re- 
search Laboratories, Air Materiel Command. 

75 (S139) 



76 (S140) ELECTROMAGNETIC WAVES 

to electric field, dv/dn = to magnetic field). They also apply to acoustic 
pressure with rigid walls {dv/dn = 0) or free walls {v = 0) . 

In formulating the problem, we attempt to represent a plane pulse incident 
on a wedge or corner. However, for certain directions of incidence the pulse is 
in contact mth the wedge at all times and thus a reflected pulse is always present. 
In the case of corners several reflected pulses may be present at aU times. In 
these cases therefore we must include the reflected pulses in the formulation of 
the initial conditions. 

In Section 2 we formulate the problem of a pulse incident on a wedge and 
state the initial conditions. In Section 3 we determine the subsequent beha\dor 
of the discontinuity surfaces. In Section 4 we introduce the method of conical 
flow and reduce the problem to the determination of a harmonic function in a 
circular sector. In Section 5 we solve for the harmonic function and thus com- 
plete the solution of the problem. In Section 6 we give the solutions for all cases 
of a pulse incident on a wedge. In Section 7 we treat the various cases of pulses 
incident in corners. In Section 8 we obtain the time-harmonic solution of the 
wedge problem as a Fourier integral of the pulse solution. In Section 9 the 
three dimensional case, in which the discontinuit}^ surface at the pulse front is 
not parallel to the edge, is considered. Section 10 is the conclusion. 

2. Formulation 

We seek a solution of the wave equation. 

(1) V,, + Vyy + V,, - -^Vtt = 

in the region <p < 6 < 2Tr — <p, where 6 is the polar angle, 6 — arg (.r + iy). 
The half-planes (walls) at = ±^ form a wedge or corner according as (p is 
less or greater than 7r/2, the case (p = 7r/2 being trivial. 

On the walls, we consider two types of boundary conditions: 

Case A: y = 0; Case B: dv/dn = 0. 

The solution which we consider will have jump discontinuities on certain moving 
surfaces, say r(x, y, z) = d. We require that r satisfy the eiconal equation [2, 3] 

(2) rl+rl + Tl = 1. 

This implies that the surface can be constructed by Huygens' principle, that it 
moves with velocity c along its normal, and that it is reflected from the walls 
according to the law of reflection. We further assvune that the reflected dis- 
continuity is plus or minus the incident discontinuity according as the boundaiy 
condition is dv/dn = or i; = 0. 

The orthogonal trajectories of a family of discontinuity surfaces S\f) are 
straight lines called rays. The set of rays through a small closed cur^'e on a 



J. B. KELLER AND A. BLANK 



(S141) 77 



discontinuity surface S(to) is called a tube. Denote by dSo the area on *S(^o) 
which spans the tube and by [vq] the jump in v across >S(^o). Let the corre- 
sponding quantities on S{t) be dS and [v]. Then we require that the magnitude 
of the discontinuity [v] vary inversely as (dSy^^, that is: 



(3) 



iSo-.o \dSo/ [v] 



Equation 3 permits [v] to be computed from [vq] on the same ray, once the dis- 
continuity surfaces are known. 

We now give the formulation of the problem for the wedge in the case for 
which no reflected pulse occurs initially and the boundary conditions A prevail. 

Problem lA: On the walls the boundary condition is z; = 0. We assume that 
initially, z; = on one side of a plane and z; = 1 on the other side, while Vt = 
everywhere. The plane front of the pulse approaches a wedge and is parallel 
to its edge, (see Figure 1). Because of the special nature of these initial condi- 
tions we also assume that the plane discontinuity is moving toward the edge, in 
order to assure a unique solution. 

It is convenient to introduce the normal to the discontinuity plane (posi- 
tive in the direction of motion), and to call it the ray direction. The angle 



Plane Pulse Incident on a Wedge 





Figure 1 



Figure 2 



between the ray direction and the x-axis we call \l/, and assume it is positive. 
The problem formulated above makes sense only ii < \f/ < 7r/2 — (p. We 
will find it necessary later to distinguish the cases < xp < cp and 

<p < i// < 7r/2 — (p. 



3. Propagation of Discontinuity 

It follows from equation (2) that a plane discontinuity surface moves 
parallel to itself ^\dth velocity c along its normal and from equation (3) that the 



78 (S142) 



ELECTROMAGNETIC WAVES 



jump or discontinuity [v] = 1 across the pulse front does not change. This 
situation continues until the time the plane reaches the edge of the wedge when 
reflected and diffracted discontinuity surfaces may originate. These surfaces can 
be obtained by Huygen's principle (a consequence of equation 2) from the con- 
figuration at the instant of contact. One finds that the incident plane progresses 
parallel to itself and that one {xp > cp) or two (yj/ < cp) reflected plane discon- 
tinuity surfaces and a circular cylindrical surface with the edge as its axis are 
produced (see Figure 2). At all later times the configuration of surfaces is 
similar to that in Figure 2 and the scale is determined by a radius of the cylinder, 
which equals ct, if ^ = is the instant of contact. 




The jimip across the original plane is unchanged, and the jump across the 
reflected plane (or planes) is its negative [v] = —1. The jiunp across the 
cylinder is zero, hoAvever, since all the rays reaching it come from the axis where 
(LSq = (see equation 3). Thus v is continuous across the cjdinder. The value 
of V everywhere outside the cylinder is known (either or 1) except in the one 



J. B. KELLER AND A. BLANK (S143) 79 

(4^ > <p) or two (\f/ < <p) ''triangular" regions bounded by the wedge, a reflected 
plane and the circle (see Figure 2). Since t; = on the wedge and behind the 
reflected plane, we assume v = everywhere in this ''triangle". This seems 
reasonable since the effect of the edge has not reached this region and therefore 
the reflection ought to be the same as from an infinite plane. We will validate 
the assumption by constructing a solution consistent with it.^ The value of v 
is now known everywhere outside the circular sector bounded by the wedge 
and the circle. Since z; = on the wedge and v is continuous across the circular 
arc the values on the boundary are known. From these values we shall be 
able to determine v within the sector. 

4. Conical Flow Method 

Since the boundary data are independent of z, we seek a solution independent 
of z. Setting, in equation (1), v^^ = 0, we obtain 

(4) y„ + Vyy 2Vtt = 0. 

c 

Let us consider the configuration of discontinuity surfaces in xyt-space. The 
circle of Figure 2 describes a characteristic cone, and the lines of that figure 
describe planes (see Figure 3) . The solution v is constant in each of the regions 
outside the cone. Clearly the boundary values of v on the cone are constant 
along each generator. The boundary conditions are also constant on the wedge. 
Thus the boundary data are constant along radial lines through the origin, 
which we locate at the vertex of the cone. We therefore seek a solution v within 
the conical sector which is constant along each radial line. 

In order to take advantage of the above assumption, we introduce special 
"polar" coordinates in xyt-spsice: 

V = [cH^ - {x' + y')r 

(5) ,=f 

6 = tan~^ y/x. 

This transformation is real within and on the cone. The surface of the cone 
is given by 

(6) p = 0, g = 00 . 
In these coordinates equation (4) becomes 

(7) (p\). + [(1 - q)val + Y^'^'^' = ^' 



^The statement about the field in the "triangular" regions need not be assumed but may- 
be proved from the other assumptions and the stipulation of conical symmetry. 



80 (S144) ELECTROMAGNETIC WAVES 

In accordance with our assumption that v is constant along radial lines, we set 

V = v{q, 6). Equation (7) now simplifies to 

(8) [(1 - 2>,L + ^-iYt)„ = 0. 

Put 

in equation (8) . This yields Laplace's equation 

The solution of (10) may be written in the form 

(11) V = ^mf(z) 

where /(a;) is an analytic function oi z = pe\ Introducing R = {x^ -{- y^Y^"^ we 
have from equations (5) and (9) 

/io\ ^ = J^ X + iy R 

U^j z - pe - ^^^ ^^3^, _ ^2y/2, P- ct + ic'f - RT" 

The cone R < ct thus is mapped into the unit circle p < 1. The problem is 
now reduced to that of finding a function analytic in an appropriate sector of 
the unit circle with prescribed imaginary part on the boundary. 

5. Solution of the Problem 

The values of v on the boundary of the circular sector in Figure 2 are, in 
the case \p < 4): 

io < P < I, e = 4> 

V = Q on \ 

V = 1 on p=l, + <^<^<27r — <^ — 6 

Tp = 1, 27r - - 6 < ^ < 27r - 

V — on \ 

[O < P <l, e - 2Tr - 4>. 

Here a = (i) — \l/,h = ^ -\- \p. 

In order to solve for v, we map the exterior of the wedge in the 2:-plane 
into the upper half of the ly-plane by the transformation: 

(13) w = re''' = {f'^zf 



J. B. KELLER AND A. BLANK (S145) 81 

where X = 7r/(27r - 24>). Thus 

(14) r = p\ 0) = X(^ - 0) = X(^ - tt) + 7r/2. 

The circular sector in which v is to be determined becomes a semicircle in 
the i^-plane with ?; = on the diameter (into which the sides of the wedge 




v-o 



Figure 4 

transform). By the reflection principle we may extend v into the whole circle, 
and obtain a boundary value problem in the unit circle. (See Figure 4; ^ = X?), 
a = \a.) 

In this case, as in all the others, we have to determine a harmonic function 

V with piecewise constant boundary values. The solution of the problem may 
be obtained as the sum of solutions which take on a specified constant value 
on one arc of the circle, the value zero on the rest. Let us write down the solu- 
tion of this special problem once for all. Suppose wa > coi (cog — coi < 27r) and 

V = c on the arc C02 > co > wi and v = elsewhere. It is not hard to show that 

V may be written in the form 

(15a) . = '- Targ { "' " exp Kco.A _ co^j^l _ 

tL[w— exp {to)i}) 2 J 

In terms of real variables we then have 



(15b) V = - arctan 

TT 



I (1 - .-) sin (^^) 

f (1 + r ) cos — - — 2r cos I co — — I 



The arctangent is taken in the interval between and t. The solution to lA 
may then be written explicitly. 



(16) 



1 ^ J -(1 - p'')sinX7r 

V ^= — 9 TOT ATI \ ^ 

^ l2p' cos X(6I + rA - tt) + (p'' + 1) cos Xtt. 

-i arctan-^ (1 - .») sm X. 



2p^ cos X(<9 - \^ - tt) - (p'^ + 1) cos Xtt 



82 (S146) ELECTROMAGNETIC WAVES 

6. Pulse Incident on Wedge in General 

In order to specify the initial and boundary conditions properly we must 
distinguish several cases: 



{1) < <p < 7r/2, < rjy < 7r/2 - <p 

(a) <rp < (p (Figure 2) 

{h) <p < yp < 7r/2 - ^ (Figure 5) 



{no initial reflected pulse) 




Figure 5 

Both cases occur if ^ < 7r/4; if ^ > 7r/4 only case a occurs. 
(2) 7r/2 — <p<\l/<Tr — (p (reflected pulse present initially) 







Fig. 6. Case 2a A. 

(a) Tr/2 - ^ <yp < ip (Figure 6) 

The solution in the diffraction region (circular sector) has the same 
form as in case la above. 

(b) <p < \p < IT — (p (Figure 7) 

The solution in the diffraction region (circular sector) has the same 
form as in case lb above. 



J. B. KELLER AND A. BLANK 



(S147) 83 



Both cases occur ii (p > 7r/4; if v? < 7r/4 only case b occurs. 

In each of these four cases we may consider either boundary condition A 
{v = 0) or B (dv/dn = 0) on the wedge. The wave fronts are shown in Figures 
2, 5, 6 and 7. In the preceding section we obtained the solution (equation 16) 







Fig. 7. Case 2b A. 



for case la mth boundary condition A (i.e., 0<;/'.<<^, t; = Oon the wedge). 
In this section we shall give the solution for lb with boundary condition A as 
well as for both problems la, lb, with boundary condition B. This will com- 
plete the solution of all the wedge problems mentioned above, since the solutions 
of cases 2a and 2b are of the same form as those of cases la and lb in the diffrac- 
tion region. The solutions are obtained just as in the preceding section. 
O^h k) (f < yp < Tr/2 — (p; Figure b] v = on wedge. 



arctan 



(1 



p'^) cos \^ 



(17) 



(1 + p'') sin \yp - 2p' sin X^J 

(1 - p'') cos X^ \ 

- 2w)) 



arctan , ^. . 

^ 1(1 + P ) sin \yp - 2p^ sin X(^ 



(la B) < ^ < <^: Figure 2; dv/dn = on wedge. 



V = I -{- - arctan 

IT 



(1 - p'') cos X(iA - tt) 



-,} 



(18) 



(1 + p'') sin \{xp - tt) - 2p' sin X(^ 
-(1 - p'') cosX(iA + tt) 



+ - arctan 

1(1 + p^^) sin \{yp + tt) 



2p^ sin X(^ - tt) 



(19) 



(lb B) (p < yp < 7r/2 — (p: Figure 5; dv/dn = on wedge. 

. = 1 - iarctanj -(1 - P^Vos X(^ - x) 

TT 1(1 + p'') sin \{i - tt) - 2p' sin X(^ - tt) 

+ 1 arctan ( -(1 - P^^) cos X(^ + ^) 

TT 1(1 + p^') sin \{yp + tt) - 2p' sin \{d - tt) 



84 (S148) ELECTROMAGNETIC WAVES 

7. Pulses Incident in Corners 

We now consider the solution of the wave equation in the region ir — <p 
< 6 < IT -}- (p where (p < t/2, and we call the region a corner. We assume 
all conditions of the problem, except the initial conditions, are the same as in 
the wedge problem. The initial conditions correspond to an incident plane 
pulse and \}/ the direction of the incident ray, satisfies (p > \J/ > 0. 

In addition to the incident plane pulse, a number of reflected pulses will be 
present initially.^ According to equation (2) the reflected discontinuity surfaces 
(or the rays orthogonal to them) satisfy the law of reflection. These plane dis- 
continuity surfaces will, by Huygens' principle, move parallel to themselves into 
the corner with velocity c, be reflected from the walls, and ultimately emerge. 
Besides the reflected plane discontinuity surfaces, Huygens' principle yields an 
additional discontinuity surface bounding a region of diffraction consistiag of 
a circular cylinder with the edge as axis. The solution will be pieceT\'ise constant 
everyAvhere except within the circular cylinder. We now consider the solution 
in this region. 

First, we must consider the multiply reflected discontinuity planes. To 
this end, we investigate the reflections of the incident rays — lines normal to the 
incident discontinuity plane. An incident ray may strike either the upper 
{d = IT — (p) or lower wall {6 = w -\- (p) first. Accordingly we designate the 
ray as of tjrpe I or type II. These types include all rays except one which strikes 
the edge first, which can be disregarded for the purpose of determining the 
reflected discontinuity planes (see Figure 10). 

The ray direction xf/^ of a ray of type I after v reflections is 

(20) xp, = (-l)^(tA + 2.ri. 

This indicates that a reflection increases \l/ by 2<^ and changes its sense. For a 
ray of type II, the ray direction (p^ after v reflections is 

(21) <p. = i-iy'\2v^ - xP). 

The total number n of reflections suffered by a ray of type I is the unique integer 
for which 

(22) TT - <p <\l/ + 2n<p < T -{■ <p. 
Thus 

If n is odd the final reflection is from the upper wall (^ = tt — ^); if n is even 
it is from the lower wall {6 = t -{- (p). 



^See Figures 8 and 9. 



J. B. KELLER AND A. BLANK 



(S149) 85 



Similarly the number m of reflections suffered by a ray of type II is given by 
(24) -2r - 2 ^ *" < ^^ + 2- 



Boundary Condition: v = o 




Figure 8 



Boundary Condition -^'^ 
6 




Figure 9 



Incident Roy Types 




Figure 10 



The last reflection is from the lower or upper wall according as m is odd or even. 
These results may be written explicitly in terms of the number theoretic 



86 (S150) 



ELECTROMAGNETIC WAVES 



function, [X], the largest integer in X. If we set X = (7r/2^), k = X - [\] we 
obtain 

^[X] + 1, for ;/. < ^{2k - 1) 

([X], for ^p > ^{2k - 1) 



[X] + 1, for i> <p{l - 2k) 



m 



M, 



for 



^ <<p{l - 2k), 



From a knowledge of the ultimate ray directions, given by equations (20) 
to (24), the ultimate reflected plane discontinuity surfaces may be determined 
since they are orthogonal to the rays. There are four kinds of configurations 
which may occur with regard to the ultimate wave fronts (i.e. planes normal to 
ultimate rays): 

Case a: m,n have the same parity and the wave fronts do not overlap. 
Case b: m,n have the same parity and the wave fronts overlap. 
Case c: m,n have opposite parity, m odd, n even. 
Case d: n odd, m even is symmetric to c. 






Cose 0) 



Case b) 



Case c) 



In addition to the ultimate plane wave fronts, Huygens' principle jdelds a 
circular cylindrical wave front with the edge as axis and tangent to the ultimate 
wave fronts at their edges. As we stated above, the solution is piece^^^se con- 




FlGURE 11 



stant everyivhere except within this circle which thus contains all the diffraction 
effects. In order to find the solution within this circle we are led by the pre- 
ceding considerations to a boundary value problem where the boundary values 
are constant on each of three segments of the arc (see Figure 11). In soMng 



m 



J. B. KELLER AND A. BLANK (S151) 87 

the boundary value problems we shall use the notation a, b, c, for the boundary 
values and ai , az , for the angles, as indicated in the diagram. 

In all these problems we map the corner into a semicircle by means of the 
transformation 

(25) w = r e'" = (z exp {^V - tt)))' 
where 

(26) ^ = i- 

In this mapping r = p^, o) = X(^ + ^ — tt) = \(d — t) -\- 7r/2. In particular, 
we set 

(27) coi = Xai 0)2 = Xq;2 . 

As we have seen, the angles oji , cog may be completely specified. To state 
the problem completely it is only necessary to find the boundary values. There 
are two cases. 

A. V = on the Walls 

The field strength is assumed to be 1 behind the initial wave front and to 
be zero ahead. The first reflected wave fronts will each add —1 to the field 
already present, the second wave fronts, +1, then —1, +1, • • • in alternation. 
Our computation of the number of reflections then tells us what the field is in 
the region ahead of both the final wave fronts, namely 



(28) Z(-l)'+ E(-l)"-l = 



(-ir' + (-!)' 



To compute the boundary conditions we simply note that if m (say) is odd the 
wave front adds — 1 to the field as it passes over, if m is even it adds + 1 (see 
Figure 8). 

B. (dv/dn) = Oon the Walls 

In this case each reflected wave front adds + 1 to the already present field. 
The region ahead of both final wave fronts therefore has the field strength 

(29) n + m - 1. 

Each of the two last wave fronts adds 1 to this value (see Figure 9) . 

The accompanying table gives all possibilities.^ 

In each of the problems we are led by the reflection principle to a boundary 
value problem on the unit circle. The circle may be divided into four arcs on 
each of which the boundary values are constant. We note in each case that 
two of the constant values are equal. The problem may then be somewhat 
simplified by subtracting this value from the boundary values. 



^Letters in parentheses refer to the cases of page 86 (Si 50). 



88 (S152) 



ELECTROMAGNETIC WAVES 



Prob- 
lem 


Parity 


Case 


Relation be- 
tween \l/n 
and <pm - 


oci 


OL1 


Boundary Values 


n m 


a 


b 


1 
c 


Al 


odd odd 


(a) 


<Pm-ypn>2Tr 


4'n + <P + 7r 


V'm + V — TT 





1 





Bl 


n-\-m 


n+m — 1 


n-f-m 


Al' 


(b) 


fm-ypn<2Tr 


<Pm + <p—7r 


lAn + ^ + Tr 





-1 





Br 


n-f-m 


n+m + 1 


n+m 


A2 


even even 


(a) 


4^n-<Pm>2l^ 


<Pm-\-f-\-Tr 


\^n + ^ — TT 





-1 





B2 


n+m 


n+m — 1 


n-|-7n 


A2' 


(b) 


4'n-(Pm<2Tr 


\pn-{-<p — Tr 


<Pm+<p-\-7r 





1 





B2' 


n-|-m 


n+w + 1 


n+m 


A3 


even odd 


(c) 


^n<(pm 


^'n + fp — TT 


<Pm-\-<p — Tr 





1 





B3 


n+m-fl 


n+m 


n-f-m-fl 


A3' 


(c) 


^n><Pm 


(pm + <p — Tr 


^n + <p — Tr 





-1 





B3' 


n + m + l 


n-\-m 


n+m-fl 


A4 


odd even 


(d) 


4'n<<Pm 


^n + <P+ir 


<pm-{'<P + Tr 





-1 





B4 


n-hm+l 


n+m 


n-l-m — 1 


A4' 


(d) 


^n><Pm 


<Pm + <P + Tr 


'An + V' + TT 





1 





B4' 


n+m-\-l 


n+m 


n-fm — 1 



i 



Let us now construct the solutions. For all the problems A set (in view 
of Equation (15b): 

1 . i a-^-)^in(^-^) I 



(7 = - arctan 

TT 



(30) 



|(1 + r^) cos (^^^) - 2r cos (co - "-^)) 



T = - arctan 

TT 



|(l+r^)cos'"^""' 



J — 2r cos (g 



+ 



CO2 + 0^1 



where coi = Xc^i and C02 = Xa:2 . 



J. B. KELLER AND A. BLANK (S153) 89 

Here the values of the arctangents are restricted to the interval between 
and X. For Al, A2', A3, A4' we have v = a - t. For Al', A2, A3', A4, v = 

T — (T. 

In the remaining problems, Bl, Bl', B2, B2^ B3, B3', B4, B4' set 



(31) 



. 1 . j (1 — r ) sm 6Ji 

^ = - arctan S 77— — 2^ 

TT 1,(1 + r ) cos coi — 2r cos w 

1 , / — (1 — r^) sincoa 

Tj = - arctan s yz — , — 2^ — r 

IT [{1 -}- r ) cos C02 — 2r cos co 



where the values of the arctangents are taken between and t. 

For Bl, B2 we have v = n-\-m-l-{-^-{-r}. For Bl', B2' we have v = 
n + m + 1 - ^ - 77. For B3, B3' we have y = n + m - ^ + 77. For B4, B4' 
we have v = n-\-m-{-^— r]. 

The solutions may easily be calculated by taking the values of ai and ag 
from the tables. Using 

^„= {2n<p-{- iA)(-ir 
(32) 

<p^ = {2mcp- rl^X-ir^' 

we find as the solution to the problems Al Al', A2, A2' 

V = -i arctan | (1 - p^^) sin Xtt 

TT 1(1 + p'^) cos Xt + 2p^ COS X((? + ^ - tt) 

(33) 

. 1 arctan | (1 - p") sin Xx | 

^ 1(1 + p'') cos Xtt - 2p' cos X(^ - ^ - 7r)J 

as solutions to the problems A3, A3', A4, A4' 



(34) 



1 ^ / (1 - p'') cos X^A 

V = - arctan \ 7; ^^-^ -^ 

^ kl + P ) sm \xp - 2p' sin X^ 



_ 1 arctan (1 - P^^) cos X^ 

^ 1(1 + p'') sin X^ - 2p' sin \{e - 27r)> 



and as solutions to all the problems B 

. = n + m + iaxctan| d - p") eos X(^ - .) 1 

T 1(1 + p"") sin X(^^ - t) - 2p' sin X(« - 7r)J 

(35) 

_ 1 arctan | (1 - p") cos X(^ + ^) ) 

T 1(1 + p") sin X(iA + tt) - 2p'' sin \{e - ir)) 



90 (S154) ELECTROMAGNETIC WAVES 

In connecting all the problems which have the same formal solution we 
have used the identity 

(36) arctan (—x) = t — arctan x 

valid when the arctangents are restricted to the interval between and tt. 

We observe that the above solutions apply also to the wedge problem if 
we permit <p to range through the entire interval < ^ < tt. In fact if we 
replace <p hy ir — (p in these equations we obtain the solutions to the wedge 
problem given previously. 

There remains the interesting question as to when there is no diffraction 
phenomenon, i.e, v is constant within the circle. This will occur under the 
following conditions for either boundary condition 

a) m, n odd; ^p^ ~ xf/^ = 27r 

(p = , — ■ independent of ^. 

"^ ^ _i_ ^ ^ ^ 

j(3) m, n even; ^n — <pm — Stt 

When X is an odd integer the emergent ray lies in the direction r — \p. 
This "mirror" property is the ordinary reflection principle for the plane, (p = t/2. 
When X is an even integer, the emergent ray lies in a direction opposite to that 
of the incident ray. This is the familiar property of the right-angled comer, 

(p = 7r/4. 



8. The Time-Harmonic Solution 

By employing Duhamel's theorem, the solution of the above problems for 
an incident plane periodic wave, or wave of any other time dependence, may 
be obtained. Thus MacDonald's solution of the wedge problem or Sommerfeld's 
solution of the half-plane (cp = 0) problem may be obtained. 

We consider the wedge problem lAa. Duhamel's theorem states that the 
time periodic solution E = v(r, 6, co) exp { —icot} may be derived from the pulse 
solution u{r, 6, t) according to the formula 

(37) v{o)) = —ioi j u{t) exp [icot] dt. 

J — a, 

Reciprocally, the pulse solution is given by 

(38) ' u{t) = -^. f ^ exp [-ii^t] die. 

Knowing the function u, we have an expression for f as a Fourier integral. 

First, Ave write the representation of u in the region ct > R hy means of 
the formula (15a), 



1 J 1 — r exp \i(o: — ir -{- (3)] 1 — r exp 

w = - < arg —- \- 7 rr^ - arg fr, 

TT 1, 1 — r exp {i{co — a)\ 1 — r exp {^( 



J. B. KELLER AND A. BLANK (S155) 91 

i{co -7r-/3)ir 

Here u is the solution of problem lAa and it will be recalled that 
R "Ix 



(39) r 



2,2 d2>1/2 



Lc« + (c^i^ - W) 



\{6 — (f), \(p = Xir — 



From the series representation 



arg (1 — re^) = — 2 ~ 



sm nx 



we have 



IT t=in 
where 

An = sin n(aj -\- a) -\- sin n(a) — a) 

(40) — sin n{o3 — ir -{- 0) — sin n(o) — x — /?) 

= 4 sin 7iX(^ — (p) sin nX(i/' — ^ + r) sin nXir. 
We seek a function v^ satisfying 



■" = [- 

let 



R 



+ (c r - /^^) 






for c^ > R. 

From Campbell and Foster [5] (909.7) we have for a, ^ > 



1 r°° JXcu^) 



I. 



2Tri ./_oo CO 

exp \ivT 



exp {— ico^) c?co 
-— sm^.cos (--jj 



< t < a 



(41) 



TV \_a \a / J ~ 



This leads us to take 



nXw 
sin hXtt 



exp { — mX7r/2 j Jn\{kR) 



92 (S156) ELECTROMAGNETIC WAVES 

where k = co/c. With this formula the full periodic solution ought to be 

00 

(42) V = 4:\ ^ exp {—in\7r/2}Jn\{kR) smn\{$ — <p) smnk{\p — (p -\- t). 

n=l 

This agrees with MacDonald's solution. It is in fact the correct solution for 
our problem, since a relatively simple computation shows that the function u in 
the region < d < R is actually given by (41). 

9. Three Dimensional Case 

If the incident discontinuity surface or pulse front is not parallel to the 
edge of the wedge or corner, it will intersect the edge at aU times. Thus reflected 
and diffracted discontinuity surfaces will be present at aU times, and must be 
included in giving the initial conditions. If these surfaces and the field are 
given correctly at the initial instant, we expect these surfaces as weU as the 
field distribution to remain geometrically congruent at all times. This, then, is 
the condition on the initial conditions. 




Equation of cone: 



Plone Pulse ot Skew Incidence 

Figure 12 



We assume that the discontinuity surfaces consist of an incident plane, two 
reflected plane sections (for a pulse incident on a wedge) and a diffracted cone 
(see Figure 12). We denote by y the angle between the incident discontinuity 
plane and the edge. It is easily seen, as in Section 3, that this configuration of 



J. B. KELLER AND A. BLANK (S157) 93 

surfaces persists with the point of intersection moving along the edge with 
velocity c/sin y. The discontinuities across these surfaces are also determined 
as in Section 3 for scalar quantities (e.g. acoustic pressure). For electromagnetic 
problems the reflected discontinuities may be obtained by employing the electro- 
magnetic boundary conditions (see [3]). In any case, the discontinuity will be 
zero across the cone and constant across the planes. 

To solve for v, we introduce the new (moving) coordinate 

(43) ^ = z-{ct/^my). 

In the a;, 2/,f -coordinates the discontinuity surfaces are stationary, and equation 
(1) becomes 

(44) v.^ + v„-^v,, = Q. 

This equation is the same as equation 4 with t replaced by f tan y. The boundary 
surfaces in Figure 12 are similar to those in Figure 3, and the boundary values 
are also constant along rays. Thus the conical flow method and solution de- 
scribed previously also apply in this case. 

10. Conclusion 

By combining the results of Luneberg on the propagation of electromagnetic 
discontinuities with Busemann's conical flow method for solving the wave 
equation, it has been possible to obtain exact, explicit, solutions of the two 
and three dimensional diffraction of pulses by wedges and corners. The results, 
in closed form, involve only elementary functions and also apply to acoustic 
problems. Since the diffracting surface must be both a cylinder and a cone in 
a:?/^-space, the only other surface which can be treated in this way is a wire of 
zero thickness and infinite length. However, the results do describe parts of 
the field resulting from diffraction of a pulse by any polyhedral surface. 

It also seems likely that the method can be extended to treat diffraction 
of a pulse by a cylinder of polygonal cross section, as well as diffraction of a 
pulse by a number of parallel wires of zero thickness (grating). In both of 
these cases, the new difficulty arises from diffraction of a diffracted wave. It 
seems that a modification of Busemann's [2] infinitesimal conical flow will suffice 
for the solution of these problems, and this investigation is already in progress. 

The major difficulty in determining the diffraction of a pulse from a poly- 
hedron is the determination of the field within the spherical discontinuity surface 
arising from a vertex as is already apparent in the case of a trihedral angle. 

By employing Duhamel's theorem, the solutions of the wedge and corner 
problems for an incident wave of arbitrary time dependence may be obtained. 
In particular, we have exhibited MacDonald's solution of the wedge problem for 
periodic time dependence. Sommerfeld's solution for the half-plane (0 = 0) 
also may be obtained in this manner. 



94 (S158) 



electromagnetic waves 
Bibliography 



1. MacDonald, H. M., Electromagneticism, G. Bell & Sons, Ltd., London, 1934, p. 79 ff. 

2. Busemann, A., Infinitesimal conical supersonic flow, Schriften der Deutschen Akademie fur 

Luftfahrforschung, Volume 7B, No. 3, 1943, pp, 105-122. 

3. Luneberg, R. K., Mathematical Theory of Optics, Brown University Lectures, 1944. 

4. Keller, J, B., Mechanics of Continuous Media, New York University Lectures, 1949-1950. 

5. Campbell, G. A., and Foster, R. M., Fourier Integrals for Practical Applications^ Van 

Nostrand, New York, 1948. 



Vector Wave Functions 



By R. D. SPENCE and C. P. WELLS 
Michigan State College 

1. Introduction 

This paper reviews the old and rather famihar problem of finding solenoidal 
solutions of the vector wave equation. Since this problem has been rather 
extensively treated for the case of spherical and cylindrical coordinates by 
Hansen [1] and others we shall confine our attention primarily to the spheroidal 
systems. 

The solutions of the scalar wave equation in these coordinates are now 
fairly well known. ^ Tables^ of both the prolate and oblate wave functions are 
available, which, though somewhat limited in extent and argument interval, 
are sufficiently extensive to make it possible to solve many interesting problems. 
Even so the scalar functions are rather complicated compared to those one 
meets in connection with the analogous problem in spherical or cylindrical co- 
ordinates. Much of the difficulty arises from the fact that the angular and 
radial spheroidal wave functions depend on the frequency as a parameter as 
well as on the usual spatial variables. As a consequence simple recursion 
formulas and relations between functions and derivatives of various orders do 
not exist. 

2. Solenoidal Solutions of the Vector Wave Equation 

We now consider the problem of constructing a set of vector wave functions 
from scalar wave functions. To assure the solenoidal nature of our solutions 
we may write the desired vector V„ in the form 

(1) V„ = V X (fW 

Paper presented at the June, 1950, Symposium on the Theory of Electromagnetic Waves, under 
the sponsorship of the Washington Square College of Arts and Science and the Institute for 
Mathematics and Mechanics of New York University and the Geophysical Research Directorate 
of the Air Force Cambridge Research Laboratories. 

^See for example Stratton, Morse, Chu, and Hutner, Elliptic Cylinder and Spheroidal 
Wave Functions, New York, John Wiley and Sons (1941). See also Leitner, A., and Spence, 
R. D., The oblate spheroidal wave Junctions, Journal of the Franklin Institute, Volume 249, 1950, 
p. 299. 

^Tables of the oblate functions are given in the paper by Leitner and Spence. One of the 
present authors (R.D.S.) has available tables of the prolate functions. 

95 (S159) 



96 (S160) ELECTROMAGNETIC WAVES 

where ^„ is a solution of 

in spheroidal coordinates. If one now inserts the assumed expression for Vn in 
the vector wave equation one finds that the unknown function f must satisfy 

(2) V^f + V(ln^S-Vf = ^ 

where the ^Pn are a set of arbitrary scalar functions. If f is required to satisfy 
the conditions (a) f is independent of i/'„ , (b) V^ f = 0, (c) f finite except at 
infinity, one finds 

(3) f = Zia or f = iT^r 

where a and r represent a constant vector and the position vector respectively 
and i^i and K^ are constants. The obvious cylindrical and spherical sjTometry 
of the two solutions for f are responsible for the remarkable simplicity of the 
vector wave functions in cylindrical and spherical coordinates. A second 
solution normal to V„ is clearly 

(4) U„ = V X V„ . 

3. Orthogonality 

We now examine the set of functions {V„} and {XJ„} for those properties 
which may be of use in solving electromagnetic problems. In cjdindrical and 
spherical coordinates one can easily show that vector wave functions are orthog- 
onal, that is 

j Wn'^mda = 0, Uy^m 

(5) j JJn'TJmda = 0, n p^ m 



I 



V„-U„ da = 



where s is an appropriate surface, provided we take f = k (a unit vector along 
the 2:-axis) for the cylindrical case and f = r in the spherical case. In general 
we have 

(6) f V„-V^ da = - j (ViAn- ViAn. - V^An-ff- VW da. 

Let us now consider the result of making a 'Svrong" choice of f for a given set 
of ypn • For example we might choose f = k and xj/n the set of spherical wave 



R. D. SPENCE AND C. P. WELLS (S161) 97 

functions. The first integral on the right in (6) vanishes and we are left with 



(7) f Yn-ymda = [ 



dz dz 



which does not vanish when integrated over the surface of a sphere. Thus we 
see that in general our wave functions are not orthogonal among themselves. 
Furthermore there is no reason to suppose that the two sets of functions {V„} 
and {U„) are orthogonal. For example if we again choose f = k and let {yj/ri} 
be either a set of spherical or spheroidal wave functions we can easily show 
that it is possible to expand certain functions in either the set {V„} or {Un). 
Thus 

i e''' = Z ^«V„ = E ^nV X kxPn 

n n 

(8) 

i e''' = X; BJJ, = X ^nV X V X ki/'^ . 

The coefficients A^ and B^ can easily be found by using the orthogonality 
of the scalar wave functions 

(9) A - - g^+l)(-l)^ B -^A 

for spherical coordinates. 

The types of solenoidal vector wave functions for which pairs of expansions 
such as indicated in (8) exist can easily be found. Let 

(10) F = V X k^ = V X V X k$ 

where ^ and $ are solutions of the scalar wave equation such that 



(11) 
where 



$ = Z hn^f^n 



(12) V^^ = Z «.VV, 



then 
(13) 



n 

* = f^ix, y)e"" + U{x, y)e-"" 

* = (lx{x, y)e"" + g^(x, y)e-"" 



sf d'g 


df_ 


d'<, 


dy dx dz ' 


dx 


dydz 



98 (S162) ELECTROMAGNETIC WAVES 

where /i , /2 , Qi and ^2 are solutions of the two dimensional Laplace equation. 
The functions of / and g are related by 

(14) 

Let us now consider certain conditions under which a set of vector wave 
functions is orthogonal. If the vector wave functions are tangent to one of 
the sets of the coordinate surfaces, sufficient conditions for the orthogonaHty 
of the vector wave functions can easily be WTitten dowTi. Let u^ , U2 , u^ repre- 
sent the coordinates, ii , i2 , is the corresponding unit vectors and hi , h2 , hs 
the corresponding metrical coefficients. The set of vector wave functions V„ 
tangent to the surface Ui = constant will be orthogonal on this surface pro^^ided 

(a) V„ = Q„(w2 , Uz)fn{ui) 
(15) 



(b) 



j;(K3F.2-F.F.2)(i2.£-i3.g)f: = 



where Vn2 and F„3 are the components of V„ along U2 and u^ and V^i and T^^g 
are the corresponding components of V^ . 

These equations are satisfied by the usual cylindrical and spherical vector 
wave functions V X )syp7^' and V X i^r^' respectively. We shall return to 
the problem of finding tangential solutions later on. 

4. Spheroidal Functions 

After the preliminary discussion it seems clear that it would be rather 
surprising to find that either choice of f yielded a set of vector wave functions 
orthogonal over the surface of a spheroid. Simple computations show that no 
orthogonality of the type (5) exists. 

To show this we make use of the fact that two solutions V;, and V^ of the 
wave equation must satisfy 

»•) I(v.Sr-'--£)jf=» 

integrated over the closed surface u^ = constant. We assume that 
(17) V„ = V X k^An or V X riAn 

and that the ypn are separable solutions of the scalar wave equation in either 
oblate or prolate spheroidal coordinates. Then i/'„ has the form F„(wi)G„(i/2 , ^^3)• 
It may be shown that (16) is incompatible with 



I 



This immediately poses the question as to just how one may hope to solve 



R. D. SPENCE AND C. P. WELLS (S163) 99 

a problem in spheroidal coordinates. The only help which seems immediate 
lies in the orthogonality of the scalar wave functions from which the vector 
wave functions are derived. In certain cases this appears to be sufficient. For 
example suppose we wish to expand 

in the functions V„ = V X Txpn and Un = V X V X T\J/r, where the \pn are oblate 
spheroidal wave functions. Since the function being expanded contains a radial 
component and since only the set {U„) can supply this radial component it is clear 
that in this case we shall not find two independent expansions as in the case 
previously mentioned. On equating components and making use of the orthog- 
onality of the scalar functions one finds 

(18) ie-''' = E CiiYiu + lUiu) 

I 

where 

Viu = V X TiUnivY'^yniO sin cp) 



(19) 



Ua. = T V X V XriunivY'^Vni^) COS^), 



k 



-i r^ 1 - cos ka{ l - v)' , s J 

^' ^ AT(i) /m; n _ 2.1/2 Un{'n) dv • 

NiVviAO)ka ^-1 U V ) 



2\l/2 

Zodd 



^' 77[sin ka(l - yy - kajl - 7}^] 



(20) 



NnVn{0)ka ^-i i — v 



I even 



A^zi = f' lunMT dv 



In the above Uimiv) represents the ^'angular" oblate spheroidal wave function, 
^^^^imi^) the first kind radial oblate spheroidal wave function, k — 27r/X and a is 
the radius of the focal circle. 

The availability of a plane wave expansion such as we have previously 
indicated leads one to consider the possibility of solving the problem of the 
diffraction of a plane wave by an oblate spheroid. If the spheroid is of non- 
vanishing thickness the boundary conditions lead to relations between the 
various components of the incident and scattered field which are quite intract- 
able in view of the lack of orthogonality of vector wave functions and the 
appearance of certain non-separable factors which prevent effective use of the 
orthogonality of the scalar wave functions. 

If the spheroid is taken of vanishing thickness the boundary conditions 



100 (S164) 



ELECTROMAGNETIC WAVES 



are much simplified. In this case, although the vector solutions are again non- 
orthogonal, one can find a formal solution of the diffraction problem. The 
solution is a dubious one however in that it leads to edge singularities of too 
high an order and seems to lack uniqueness. These questions have been dis- 
cussed by Boukamp and Meixner [2] but the exact nature of the diflSculties does 
not appear to have been completely settled. This is rather surprising in \4ew 
of the fact that singularities of the proper order and of the same order expected 
in the electromagnetic problem appear quite naturally in the corresponding 
scalar diffraction problem. 

5. Simple Boundary Problems 

We now turn to two problems which are closely related — the free oscillations 
of a spheroidal conductor and the oscillations of a spheroidal cai'ity. The 
problems differ only in that in the first case one uses the exterior or third kind 
radial functions while in the second case one must use the interior or first, kind 
radial functions. The resonant frequencies of the cavity problem are of course 
real w^hile those of the conductor problem are necessarily complex vrith. the 
imaginary part representing the radiation damping. 

To be definite let us again fix our attention on the oblate case. Consider 
the solution V X T\f/im where \J/im is a solution of the scalar wave equation in 
oblate spheroidal coordinates. One has 



V XT^Pl^ 



= [".( 



-^ 



(21) 



[{^ + ^^)(i - v')Y'' 



/a^ + ^^[(f + '7^)(iVf)rva^ 



cosj 



where the unit vectors 771 , ^i and §1 have the directions indicated in the figure 
below. 




Fig. 1. The unit vectors for the oblate spheroid. 



Let us consider the problem of trying to satisfy the boundary conditions 
of the cavity resonator or free oscillations on the spheroidal surface ^ = ^0 • 



R. D. SPENCE AND C. P. WELLS (S165) 101 

If we assume that V X Txpi^n represents the electric field then the boundary 
condition requires the tangential rj and (p components to vanish. Setting 
Vimi^o) = satisfies this condition for the rj component. But to fulfill the 
condition on the (p component we must also set d/d^ Vimi^o) = and this condi- 
tion cannot be met by the radial functions. If on the other hand we consider 

V X Tipim to represent the magnetic field we again find that both Vi^{^) and 
its normal derivative must vanish. Similar statements may be made about 

V X V Xrxl^im . 

Part of the difficulty arises from the fact that both V X r\l/im and its curl 
contain all components of the field. In the spherical and cylindrical cases 
Maxwell's equations yield two distinct types of solutions — the TE and TM 
modes in which either the electric or magnetic component in the direction of 
propagation is absent. This happens in the spheroidal cases only in two 
special instances. The first is that in which variable ^ is zero and the second 
is the one in which the field is independent of the azimuthal variable (p. In 
the case of the limiting spheroid, ^ = 0, because the radius vector is tangent 
to the spheroid and in the rj or negative r] directions, V X Txpi^ contains only 
the ^ and (p components, while V X V X ri/'i^ presents the full complement of 
components. 

The cavity resonator problem obviously has no meaning for a spheroid of 
zero thickness but we may still discuss the free oscillation of a spheroidal con- 
ductor of zero thickness. By setting either f/^(0) or id/d^) Vir„{0) to zero we 
can make V X ryj/i^ satisfy boundary conditions of either the magnetic or 
electric type. As yet we have not investigated such oscillations in detail. Such 
an investigation should enable one to predict the resonance frequencies of the 
scattering cross section in the diffraction problem. In connection with the 
thin disk problem we may also consider solutions of the form V X ^-^im • One 
has 



(22) 



d(p 



+ ^\ ^^^^ }\n Tn - ^ ^j>-('')''".tt) cos/"^ 



If we try to use these fields to satisfy boundary conditions on a spheroid 
of finite size we find the same difficulty as previously mentioned, i.e.: both 
^ztn(^o) and d/d^ Vi^(^o) are required to be zero. If, however, the field has azi- 
muthal symmetry or if we deal with limiting spheroids we may again satisfy 
boundary conditions of the electric or magnetic type. 

Solutions having azimuthal symmetry are formed by adding two non- 
solenoidal solutions of the vector wave equation: 

(23) Vi = -i sin (pUn(r])Vn{^) + j COS <pUn{rj)Vn(0 = hUiiivJ^ni^) . 



102 (S166) ELECTROMAGNETIC WAVES 

The result is clearly solenoidal. A second solution is obtained by taking the 
curl of this solution. One finds in the oblate system 

(24) 



-i¥+%-4,^^'-^'y''-'^^4 



These wave functions have rather useful properties. We note that not all the 
components appear in both solutions which simplifies the boundary value prob- 
lem. Although they are not individually orthogonal in respect to integration 
over a spheroidal surface, one can make good use of the orthogonality of the 
scalar wave functions. The magnetic and electric boundary conditions of the 
free oscillations of the spheroid or of a spheroidal cavity resonator can be simply 
satisfied by setting either f,i(^o) or its derivative to zero. 

The analog of the above fields in the prolate system has been the starting 
point of many investigations of the antenna problems.^ Actually they suffice 
only to discuss the free oscillations of the antenna or a symmetrically excited 
transmitting antenna. In the case of a receiving antenna of non-vanishing 
thickness with its axis parallel to the electric vector the excitation is onh^ ap- 
proximately symmetric about the axis but the largest terms in the scattered 
field expansion are certainly the symmetric ones and one makes no great error 
by neglecting fields of higher symmetry when imposing the boundary conditions. 

In connection with the oblate case one has a quite different group of prob- 
lems which can profitably employ the symmetric field. One of these which 
has been solved recently [3] is that of an antenna on the axis of a circular disk 
which represents a finite ground plane. This problem has a rather special 
interest as the actual calculations were done for an oblate spheroid of vanishing 
thickness. The edge singularities which are currently blamed for the difficulties 
of the plane wave diffraction problem failed to give difficulty. 

Other problems which appear quite feasible in the oblate case but which 
have not yet been worked out in any detail are the diffraction of a circularly 
symmetric wave by an aperture, the radiation of flanged transmission lines and 
waveguides operating in circularly symmetric modes, and the circularly sj^m- 
metric modes of the hyperbolic horn. 

At this point it seems clear that the vector wave functions of the spheroidal 
systems are really satisfactory only for the description of circularly s^onmetric 
fields. While it may well be that the oblate wave functions ^^ill prove of value 
in discussing the diffraction of plane waves by a thin disc this seems by no 
means assured. We have contented ourselves ^\dth discussing the more or less 
conventional types of solutions thus far since we have felt that an examination 
of these was of first importance. 

^See for example Page, L,, The electrical oscillations of a prolate spheroid. Paper II. 
Prolate spheroidal wave functions, Physical Review, Volume 65, 1944, p. 98. 



R. D. SPENCE AND C. P. WELLS (S167) 103 

6. Existence of Tangential Solutions 

We now consider the problems of trying to find other solutions. One of 
the difficulties we have previously mentioned is that except in the circularly 
symmetric case neither of the solutions are tangential to the spheroids. The 
question arises as to whether it might be possible to find by methods quite 
different than those indicated here, solutions which have the tangential property. 
The answer to this question appears to be negative. If one assumes that it is 
possible to have a field in which the component normal to the spheroid vanishes 
and then writes down the equations which the remaining components must 
satisfy one is led to differential equations which have no solution. 

Let P = (0, P2 , ^3), be a solution whose component normal to the spheroid 
Ui = const., is zero. Then if we require that P be solenoidal, we must have 

(25) V-P = and V X V X P - A;'P = 0. 

If these equations are written in component form it can be shown that Pg must 
satisfy the differential equations* 

(26) Mil) ^ _ OzUfl) ^ + 2P. = 

Ml 3mi U2 dUi 



(27) 

+ \m + ul) + M§if-=LM+i)1p^ = 0. 
L {ui + U2){1 — U2) J 

From (26), we see that Pa must have the form P2 = {^/x)f{x/y)g{us) where 
X = ul + 1, y = 1 — ul , and/ is an arbitrary function. By direct substitution 
it can be seen that (27) has no solution of this form except in the cases (a) A; = 0, 
(b) X OT y = const. Hence, in general no tangential solutions exist for the 
spheroid. 

Case (b) implies a solution exists which is tangential to a given constant 
spheroid. 

The procedure outlined above should lead to a general criterion for those 
surfaces which possess tangential solutions. We do not have the complete 
answer to this as yet. One set of sufficient conditions for such surfaces is that 
the metric coefficients must satisfy hi — hi{Ui), hz/h^ = g{u2). These conditions 
are, of course, satisfied by spherical surfaces. However, we have not constructed 
solutions for other cases. Assuming that it might be possible to find such 
solutions it is still doubtful whether they would be orthogonal over the particular 
surface to which they are tangent. 



^These calculations are based on the oblate spheroid. Similar results can be obtained for 
the prolate case. 



104 (S168) ELECTROMAGNETIC WAVES 

There is one further possibiHty that should be mentioned. It consists of 
representing the field in rectangular components with each of the components 
an infinite series of scalar wave functions. Such a representation has the con- 
siderable advantage that it allows one to make excellent use of the orthogonality 
of the scalar wave functions. In actual practice it is rather difficult to force 
the solenoidal requirement on such a representation. Moglich [4] has attempted 
to solve the diffraction problem by this method but it appears that his solution 
has the wrong type of edge singularity. 



Bibliography 

1. Hansen, W. W., A new type of expansion in radiation problems, Physical Review, Volume 47, 

1935, p. 139. Hansen, W. W., and Beckerley, J. G., Radiation from an aerial over a 
plane earth of arbitrary characteristics, Physics, Volume 7, 1936, p. 220. Hansen, 
W. W., Transformations useful in certain antenna calculations, Journal of Applied 
Physics, Volume 8, 1937, p. 282. 

2. Bouwkamp, C. J., A note on singularities occurring at sharp edges in electromagnetic diffraction 

theory, Physica, Volume 12, 1946, p. 467. Meixner, J., and Andrejewski, W., Strenge 
Theorie der Beugung ebener electromagnetischer Wellen an der vollkommen leitenden 
Kreisscheihe und an der kreisformigen Offnung im vollkommen leitenden ebenen Schirm, 
Annalen der Physik, Volume 7, 1950, p. 157. 

3. Leitner, A., and Spence, R. D., Effect of a circular groundplane on antenna radiation, Journal 

of Applied Physics, Volume 21, 1950, p. 1001. 

4. Moglich, F., Beugungserscheinungen an Korpern von ellipsoidischer Gestalt, Annalen der 

Physik, Volume 83, 1927, p. 609. 



The W.K.B. Approximation as the First Term of a 
Geometric-Optical Series 

By H. BREMMER 

Philips Research Laboratories, N. V. Philips' Gloeilampenfabrieken, Eindhoven, Netherlands 

1. Introduction 

Applications of the W.K.B. approximation usually refer to the propagation 
of waves through an in homogeneous medium. With respect to such problems 
it is often possible to interpret this approximation as the first term of an infinite 
series, each term of which represents waves that are produced by a particular 
number of reflections inside the medium/ In this paper we shall investigate 
this series with reference to the simplest application of the W.K.B. approxima- 
tion. This application concerns the ordinary differential equation: 

(1) (d'y/dx') + k\x)y = 0. 

In scalar optics this equation describes the behaviour of a plane wave 
propagated perpendicularly to the stratifications of a medium whose refractive 
index /z(a:) = k(x)/ko depends exclusively on the coordinate x. The quantity 
k{x) can be interpreted as the local value of the wave number 27r/X(a;). Equa- 
tions of the type (1) may also occur in vectorial problems; e.g., (1) is satisfied 
by the amplitude of a Hertzian vector in the case of a spherically symmetric 
medium with variable dielectric constant and a magnetic permeability equal to 
unity.^ 

2. Derivation of the W.K,B, Approximation of (1) from a 
Discontinuous Model 

In what follows we consider an inhomogeneous space re > (with variable 
wave number k{x)) that is adjacent to a homogeneous space x < ^\'ith a con- 
stant wave number /cq . Provisionally we replace the inhomogeneous space 

Paper presented at the June, 1950, Symposium on the Theory of Electromagnetic Waves, under 
the sponsorship of the Washington Square College of Arts and Science and the Institute for 
Mathematics and Mechanics of New York University and the Geophysical Research Directorate 
of the Air Force Cambridge Research Laboratories. 

^See H. Bremmer, Handelingen, Natuur-en Geneeskundig Congres, Nijmegen 1939, p.; 88 
Philips Research Reports 4, p. 189, 1949; Physica, Volume 15, p. 593, 1949. The one-dimen- 
sional problem has also been attacked by R. Landauer in his thesis. 

^Compare H. Bremmer, Terrestrial Radio Waves, Elsevier Publishing Co., Houston- 
Amsterdam, 1949, p. 138. 

105 (S169) 



106 (S170) 



ELECTROMAGNETIC WAVES 



X > by a set of homogeneous layers < x < Xi , Xi < x < X2 , X2 < x < x^ , 
• • • with the successive constant wave numbers ki , kz ^ ks ^ • • • (see Figure 1). 



XsAh 



X=Xs 



4Xn 


Pn 


\Tn+l 




|^^s-h7 


Ps.l 


]rs.2 i 


pXs 


Ps 


>s-.; 1 



A-n 



^s+; 




Later on we pass from this discontinuous medium to a continuously changing 
one by making the thicknesses Ax^ = x^ — x^-i oi the various layers infinitely 
small. 

We consider a plane wave exp {i(koX — mt) } arriving from the space x < 
and travelling in the direction of increasing x] for convenience we term this 
direction ''upwards," that of decreasing x ''downwards." We shall investigate 
the behaviour of the wave exp {ikox] (the time factor exp {—iwt} T\'iU be 
omitted in the following discussions) when it enters into the discontinuous 
medium above the level x > 0. At the boundary a; = of the first layer the 
wave exp [ikax] is split into (-?) a refracted wave Pi penetrating into this layer 
and represented by Dq exp [ik^x], {2) a reflected wave Ty_ returning to the 
space re < and represented by Rq exp [—ikox]. The coefficients Do and Rq 
are to be derived from the boundary conditions at x = 0. At any boundary 
we assume that u and du/dx shall be continuous. For the splitting of exp {ikox] 
into the waves Pi and Ti at a: = these conditions lead to the formulae: 

Ro = {h - k,)/iko + A;i); Do = 2ko/(ko + k,). 

The wave Pi will be split at the next boundary x = X2 into a refracted wave 
P2 penetrating into the second layer and a refracted wave T2 returning to the 



I 



H. BREMMER (S171) 107 

first layer; these waves are proportional to exp {ik2x} and exp {—ikix} re- 
spectively. This procedure of splitting is repeated at each next boundary. The 
boundary conditions for an arbitrary level x = x^ lead to the following ratio of 
the amplitudes of P, and P^+i at this level: 

(2) P,,,{x,)/PXxs) = 2kJ{K + A:.,0. 

The chain of waves consisting of the sequence Pi , P2 , P3 , • • • may be 
termed the principal wave P. With the aid of (2) and the proportionality of 
this wave to exp {iksx} in the 5-th layer we easily derive for the value of Uo 
just below the level x = x^ : 

7. , \ exp {iki Axi\ , ' exp {iki AX2] • • • 7 XIT ^^P !^*^«^^«} • 

With a view to the transition to a continuous medium, to be performed next, 
we write the latter expression in the alternative form: 



(3) Uoixr^ - 0) = exp <^ - X; log (1 + Aks/2k.) + i E ^.Ax. \, 

in which we have introduced the finite differences Ak, = k^+x — k^ for the wave 
number. Passing to a continuous medium {AXs — > 0), the second sum in the 
exponent of (3) is transformed into the integral i /'=o" k{s) ds, the first sum into 



/ 



s=x, 



dk, _ _1 , k(Xn) 
.=0 2K ~ 2 ^""^ k{0) ' 



Thus we obtain the following formula for the principal wave in the continuous 
case: 

Uo(x) = exp ^ — - log (k{x)/ko) + W Hs) ds?, 
or 
(4) Uo{x) = {j^y exp \i £ k{s) ds\. 

This expression just represents the W.K.B. approximation for the wave 
produced in the inhomogeneous space by the primary wave exp {ikox} arriving 
from the homogeneous space a: < 0. Consequently we can give the following 
interpretation to this W.K.B. approximation: it represents the wave originating, 
by refractions, directly from the primary wave which arrives from the adjacent 
homogeneous space; its intensity is determined by the reflection processes 
which take place in any infinitely thin layer of the inhomogeneous medium. 

3. The First Correction Term to the W,K.B. Approximation 

The above derivation of the W.K.B. approximation suggests how to get 
additional contributions to the wave function. Returning to the discontinuous 



108 (S172) ELECTROMAGNETIC WAVES 

model of Figure 1 we consider the reflected waves Ti , T2 , T^ , • • • that are 
generated by the principal P wave. These T-waves are travelling downwards, 
and cross the set of boundaries a; = a:, in a direction opposite to that of the P- 
wave. Each T-wave again produces a reflected wave (being a rising wave in 
this case) at any boundary. Owing to the inversion of the direction of propaga- 
tion, the amplitude of a T-wave is multiplied by a factor 2/uVxi/rAv-i + k^) 
when crossing the level x = x^ instead of the factor 2k^/{k^+i + k^) applying 
to the P-wave. 

Let us consider the special wave T^+i generated at the boundary- x = x, . 
This T-wave starts with an amplitude determined by the local value Uo{x^ — 0) 
of the P-wave and by the reflection coefficient referring to a rising wave crossing 
the level x^ . Thus we obtain the following initial value of T^+i : 

(5) T,.,(x. - 0) = uoix, -0)^' ~ ^'^' 



kg 4" kg 



The modifications undergone by jT^+i when travelling from x = x, to the 
lower level x = Xm are of the same type as those encountered b}- the rising 
P-wave. Thus, by analogy to formula (3), we get the following ratio of the 
amplitudes of Ts at the levels mentioned: 



T^^^ixm + 0) 
Tg^.ix, - 0) 




= expj- 


- Z log (1 

<T=m + 2 



(6) ^ 

Ak,.J2kJ) + i S k,Ax,y x„ < x, 

a=m+\ ) 

The multiplication of (5) and (6) leads to a formula for T^+x{Xm + 0) in 
which the transition to a continuous medium may be very easil}- performed 
once more. The reflection coefficient occurring in (5), accordingly, is trans- 
formed into —dkj2k, . In this way we arrive at the following amplitude at 
the level x for the wave that has been split off by reflection from the principal 
wave inside the infinitely thin layer s < x < s -\- ds\ 

-uois) ^ exp |- j^__^ J^ + ^j^ m da 

= -2mr Mr '^' '^^ i' L ^^"^ ^' 

All the layers for which s > x, provide a similar contribution to the total 
wave u, which includes all the T-waves spHt off from the principal wa^'e after 
one single reflection. The connection between this wave Hx and the original 
P-Avave Uo thus proves to be given by 



H. BREMMER 



(S173) 109 



4. General Terms of the Complete Geometric-Optical Series; 
Recurrence Relations. 

As remarked in the preceding section each wave T^ generates a reflected 
wave (travelhng upwards) when crossing any of the discontinuity levels of the 
model of Figure 1. These new reflected waves, [/-waves, are thus produced 
after two successive reflections, one connected with the generation of the U- 
wave from T, , the other one connected with the generation of the Tg-wave 
from the original P-wave. In their turn the [/"-waves produce further reflected 
waves which are thus produced after three various reflections. This reflection 
procedure is repeated ad infinitum. Therefore we get a complicated pattern of 
rays each of which is produced as the result of a definite number of reflections. 
Examples of such rays are shown in Figure 2 in which the number A^ attached 
to each ray refers to the number of reflections needed for the production of this 
very ray. 







N=2 








A/=/ 


■ 


N^3 


N=0 












N=3 


A/=4 


N=0 






Nzz2 










A/=; 





















Figure 2 



The A^-classification here described still holds when passing to the limit 
of a continuously changing medium. For the latter we indicate by U^ the 
common contribution of all the waves produced by one and the same number 
of reflections N, the reflections in this case taking place in infinitely thin layers. 
With a view to this iV-notation we have already marked the principal wave P 
by Uq because this wave is produced without any reflection at all. Furthermore 
the system of the T-waves (which originate after one reflection) was accordingly 
marked Ui in formula (7). Evidently all the terms U2n with even subscripts 
2A^ represent up-going waves, the terms ?i2iv+i with odd subscripts down-going 
waves. 

The explicit expression for an arbitrary u,y is complicated while recurrence 



110 (SI 74) ELECTROMAGNETIC WAVES 

formulae connecting two successive t^ivrterms are rather simple. In this respect 
we recall that the derivation of u^ in (7) is independent of the anahnical form 
of the function Uo(x). The same derivation therefore holds for the relation 
between any odd term U2N+1 and the preceding even term Uzx , the former being 
the result of the reflection losses of the latter. A similar relation can be derived 
for the dependence of the up-going i^ai^-wave on the preceding do^Ti-going U2X-1- 
wave. The two recurrence relations imder consideration read explicitly: 

(8a) U2n{x) = 2[k(x)Y^^ Jo ^^ [k(s)Y^^ U2n-i{s) exp <i J kio) daj, 

1 r°° k\s) / f 

(8b) U2N+iix) = - 2[Ux)Y' ^ J ^^ JkisJY^ '^^''^^^ ^^^ T J ^^^""^ ^"^ 

5. The Complete Series ^ u^ as a Solution of the Differential Equation 

In the discontinuous model the series 2Z^=o u^ix) represents the complete 
solution corresponding to the primary wave exp { i koX ] arriving from the homo- 
geneous space a; < 0. As a matter of fact we are sure not to have omitted 
anything at all when adding the contributions of all the rays that are possibly 
generated by any number of refractions and reflections at the discontinuity 
boundaries. This suggests that the same ^\dll hold in the limiting case of the 
continuous medium. We then have to verify that the differential equation (1) 
is satisfied by the series y = 2Z^=o Un , the recurrence relations (8) being given. 

This verification is performed as follows: We start by deriving new relations 
from a differentiation of (8) with respect to x, viz., 



(9a) u'2N = "(2^ ~ ikju2ii + -^ 



U2N- 



(9b) u'2N 






Another differentiation of any of these identities, combined with an apph- 
cation of the other identity, leads to the follomng additional relation containing 
three consecutive terms 

(10) ^ + ^ «- = ii 1^ - Ykh + \2k- i^n-^ + w "-= • 

It is remarkable that here the discrimination between even and odd values 
of N has disappeared. The next step concerns the summation of (10) over 
A^ = 2, 3, 4, • • • while substituting 

J^'^N = y; J2'^N = y — uo ; ^u^ = y — wo — ih • 

^■ = N=l N = 2 



H. BREMMER (S175) 111 

The terms with Ui can be expressed in Uq with the aid of (9b) for iV = 
and of the differential quotient of this relation. The result of the summation 
mentioned then proves to be 

dx'^^^ ~ dx' '^ 2kdx^V 2" 2k' + 2kr ' 

The right-hand member appears to be zero when evaluated according to 
(4). The series y = ^n=q Un (if the series for y" is convergent) has thus been 
verified as a solution of the differential equation (1) in so far as the summation 
of (10) over A^ be legitimate (an example is given in section 8). The W.K.B. 
approximation now appears as the first term of an infinite series each term of 
which can be interpreted geometric-optically by the number of reflections 
needed for the production of the contributions represented by that term. 

6. Relations for the Total Rising and Downgoing Wave 

As remarked before, the terms Un with even subscripts correspond to rising 
waves, those with odd subscripts to downgoing waves. Accordingly, we can 
split the complete solution y = u into two parts, the total upgoing wave 

(11a) Wt = Uo -\- eU2 + eV + ' * ' , 

and the total do^vngoing wave 

(lib) Ui = €Ui + €^2/3 + €^^^5 + • • • , 

in which the parameter e has to be taken equal to unity. This parameter is 
here introduced as a convenient expedient for the derivation of the several 
2/jv-terms, as will be clear from the next sections. Evidently the power of e, 
like the subscript A^, indicates the number of reflections involved in the term 
under consideration. 

The total up-going wave Wt ^.nd the total downgoing wave u^ satisfy a 
set of integral equations which lead to a rather simple survey of the consecutive 
i/^-terms. These integral equations are obtained at once from (8a) by multi- 
phcation by e^^ and a summation over N = 1, 2, 3, • • • and from (8b) by multi- 
plication by e^^"^^ and a summation over A^ = 0, 1, 2, • • • . The equations in 
question are then found to be 



(12a) u^{x) - 2f^ZTjT72 j ds T^T^yp ^i(s) exp U J k{a) da? = Uo(x), 
(12b) u,ix) + ^j^T7. I ds j^|k u,(s) exp [i I k(a) daj = 0. 



These integral equations are completely equivalent to the relations (8) 



112 (SI 76) ELECTROMAGNETIC WAVES 

because the latter can be obtained from (12) by equating to zero the terms 
occurring with one and the same power of e after substituting (11). 

It is also possible to eliminate either of the functions Uf{x) or Ui(x) from 
(12) in order to obtain a single integral equation of the Volterra tyipe for the 
other function. The development oi u-^ or u^ into powers of e proves to be 
identical with the Neumann-Liouville expansion of the solution of any of these 
single integral equations with the aid of iterated kernels. Thus our geometric- 
optical splitting can be reduced mathematically to a well-kno^\Ti method for 
solving integral equations. 

We conclude this section with the derivation of some other relations con- 
cerning the functions u-^ and Ui . For this purpose we multiply (9a) by e^"^ 
and make a summation over A^ = 1, 2, 3 • • • and also multiply (9b) by €""^"^ 
while summing over iV = 0, 1, 2 • • • . An addition of the resulting identities, 
using moreover the relation u = u^ -\- Ui , leads to the relations in question: 

^ «' + (!-. )|« 



Wt = o + 
(13) 



2ik 



""' - 2 2ik 

We recall the introduction of e as an expedient for the derivation of rela- 
tions between the u^ terms. The actual value e = 1 reduces (13) to the simple 
formulae 



(14) 






u w 
Ui = - — 



2 2ik' 



The influence of the inhomogeneity of the medium on the appHcabilitj^ of 
the W.K.B. approximation is shown very clearly by finally deriving the fol- 
lowing identities from a differentiation of (14) and an application of (1): 



dx 

(15) 

dx 



^ ]k''' exp [-if k{s) ds\u,{x)j =^X k''' exp l-i f kis) ds\u,(x), 
\k''' exp li f k(s) ds\u,{x)\ = 1^ X k''' exp li j k{s) ds\u,{x). 



In fact, a small inhomogeneity indicates small values of k' and a possible 
neglection of the right-hand members which leads to the W.K.B. approximation 
for u^ and u^ . The deviation from the W.K.B. values is directly dependent 
on the numerical values of the right-hand members of (15). 



H. BREMMER (S177) 113 

7. Transition to Other Variables instead of u and x 

The determination of the w^-terms is considerably facihtated by the intro- 
duction of the quantities 

^ = r k(s) ds and lix) = [k{x)y'\{x) 

instead of x and u respectively. The terms of the series I{x) = ^n^q e^I^ix), 
which corresponds to the original series u(x) = X]^=o e^u^ix) are then simply 
obtained as follows: the solution of the differential equation 

(16) S + {' ~ ' ^ " '^'^^®}^ = ° 

which satisfies the boundary conditions 

7(i + ieR) - i% = 2(ko/'' at ^ = 0, 



(17) 

d^ 



7(1 — ieR) + ^ — = at infinity, 



is developed wit^ respect to e (the coefficient of e^ yielding I^) ,* the new param- 
eter R is defined by 

P _ dk /dx _ dk/d^ 
^ ~ 2k' ~ 2k ' 

The correctness of this procedure is demonstrated as follows: The differ- 
ential equation (16) is derived by multiplying (10) by e^, by summing over 
A^ = 2, 3, • • • and final transition from the variables u and a: to 7 and ^. Further, 
in virtue of (13) the transposition of the boundary conditions (17) to the original 
variables u and x, simply reads 

(18a) u^ = Uo = 1 at a: = 0, 

(18b) Ui = at x =oo. 

The first condition (18a) states the vanishing at a; = of the difference of 
the total rising wave and the principal wave Uq . The contributions to this 
difference, as observed at a special level x, are generated in the space between 
the x level under consideration and the boundary at a; = 0; this difference 
therefore vanishes if the space mentioned is reduced to a zero thickness, i.e. 
if the X level approaches the boundary x = 0. The other condition (18b) is 
verified by the corresponding property at infinity, namely the vanishing there 
of the total downgoing wave. 



114 (S178) 



ELECTROMAGNETIC WAVES 



It is to be noted that the variable ^ is proportional to the number of wave- 
lengths 



Lw) = iL^^'^'^ 



comprised between x = and the level under consideration. In other words, 
the introduction of ^ amounts to taking the local values of the wavelengths as 
units when measuring the distances x. Finally the quantity R(x) can be in- 
terpreted as the reflection coefficient of a layer with thickness l/k{x) = X(a:)/2T. 



8. An Example of the Geometric-Optical Series 

The preceding theory may be illustrated by the inhomogeneous medium 
with a constant value — i^o of RiO- The equation (16) here reduces to 

~ + {l- e'Rl)I = 0. 

The solution satisfying the boundary conditions (17) reads 
1 - ieRo - (1 - e'RlY^' 



(19) I = liko) 



eRo 



exp {z«l - e'Riy^'] ^ > 0, 

if we assume ^ complex with a small positive argument so that e' ^ -^ and e~ -^co 
at infinity (slightly absorbing medium). The function R{^) = —Ro corresponds 
to the situation: 







ko , 




X <0 


U + 2koRox' 


x> 0, 


which is shown schematically in Figure 3. 




k 


W 






^ 










e/Abx 


\ 






fRoe-'^^ 
iR§ e-'V- 




lZ] \ 








.^- 


\. 












^" 













FiGUl 


iE 3 





The primary wave exp {ikox} arriving from the homogeneous space x < 
produces reflected waves in the inhomogeneous space x > 0. These reflected 



H. BREMMER (S179) 115 

waves reach the homogeneous space in which we can represent their sum by a 
function of the form u{x) = T exp {—ikox}. The quantity T acts as the re- 
sulting reflection coefficient of the complete inhomogeneous space extending 
from X = to a; = c». The ^^- value of the total field at a; = — , viz. 1 -\r T, 
must be equal to the u-value at a; = 0+, viz. 7(0) /(A^o)^^^- In virtue of (19) 
this boundary condition leads to the follo^\ing value of T: 

„ . 1 - (1 - ^Rir' 

1=1 5 . 



The expansion 



T=-iti-i)fl,'yR.r-' 



shows the distribution of the amplitudes of the total waves that are generated 
by 1, 3, 5, • • • reflections in the inhomogeneous space. Indeed, according to 
the above theory these amplitudes are equal to the coefficients of e\ e^, e^ • • • . 
The first few reflected waves are thus given by the following expressions 

1 reflection: iy jRo exp {—ikox} = - Rq exp {—ik^x}, 

2 reflections: —iy ' te exp [—ikox] = - Rl exp [—ikox], 

3 reflections: ii jRl exp {—ikox} = —Rl exp {—ikox}. 

The amplitudes of the waves reflected to the homogeneous space are thus 
the binomial coefficients of J in this example. At the same time we infer the 
convergence of the splitting procedure if eRo = Ro < 1. 

As for the convergence of the ^Ar-series in other examples, the well-known 
insufficiency of the W.K.B. approximation in the neighbourhood of zeros of k{x) 
can be interpreted in our theory as a very slow convergence of the Uu-series. 



Remarks Concerning Wave Propagation 
in Stratified Media 

By S. A. SCHELKUNOFF 
Bell Telephone Laboratories 

1. Wave Equations 

Under a great variety of conditions the problem of wave propagation re- 
duces to one or more equations of the form 

(1) g + F(a.)« = 0. 

The complete wave function is u exp {ioit}, where t is the time and co is the 
frequency in radians per second. If F{x) is not analytic, u and its first derivative 
must be continuous. In the case of uniform plane waves of frequency o, incident 
on a horizontally stratified layer, (Figure 1) u is either the electric intensity E 

(3) 



A « 




Fig. 1. A uniform plane wave incident on an inhomogeneous layer. 

or the magnetic intensity H according as E or H is parallel to the layer. If 
the angle of elevation A is small, 

(2) F(x) = co^M«,[2(« - n.) + A^], 

where n, e, n are respectively the permeabihty, the dielectric constant and the 
index of refraction. The subscripts refer to the homogeneous medium below the 
stratified medium. It is assumed that /x is constant throughout the entire medium. 
Equation (1) is not the primary wave equation in the sense that, in general, 
it does not arise directly from the physical laws and does not express the physical 

Paper presented at the June, 1950, Symposium on the Theory of Electromagnetic Waves, under 
the sponsorship of the Washington Square College of Arts and Science and the Institute for 
Mathematics and Mechanics of New York University and the Geophysical Research Directorate 
of the Air Force Cambridge Research Laboratories. 

117 (S181) 



118 (S182) ELECTROMAGNETIC WAVES 

conditions completely. For example, if E is parallel to the layer and Ht is the 
component of H parallel to the layer, we obtain 

(3) ^= -ic^fxH, , ^ = -ic^eMn - n,) + A']E, 

Since /x is independent of x, we may reduce (3) to (1) by eliminating Ht . In 
the more general case we have a system of two first order wave equations with 
two wave functions 

(4) ^ = -ifix)v, I = -ig{x)u. 

If either f(x) or g{x) is not analytic, u and v are required to be continuous. If 
either f{x) or g{x) is independent of x, then either u or v satisfies (1) and the 
corresponding boundary conditions. But even in this case there are some 
advantages in deahng directly with (4) rather than wdth (1). In any case, the 
proper definition of the reflection coefficient requires both wave functions u, v, 
even though one of them may be the derivative of the other. 

2. Definition of the Reflection Coefficient 

In a homogeneous medium f{x) and g{x) are constants and the general 
solution may be expressed as the sum of two progressive wave functions 

(5) t* = Ae-'^'' + Be'^'% v = K-,\Ae-'^'' - Be'^''), 
where 

(6) ^, = (/,?o"^ K, = (/,^o"^ 

The parameter Ki equals the ratio u/v for the wave traveling in the positive x 
direction and — u/v for the wave traveling in the opposite direction. 

If the source of the wave is at a; = — «» and if the medium is discontinuous 
at a; = 0, the ratio q = B/A is called the coefficient of reflection for the u- 
f unction. Expressing q in terms of u{0) and v(0), we have 

Thus, the reflection coefficient depends solely on the ratio of the wave functions 
at the discontinuity and on the parameter K, of the homogeneous medium 
containing the incident wave. 

In general, no meaning can be attached to the reflection coefficient when 
the incident wave is in an inhomogeneous medium for the simple reason that 
in general we cannot decompose the total wave into ''progressive" components. 
A formal expression such as 

(8) u= A{x) e"*"\ 



S. A. SCHELKUNOFF (S183) 119 

in the form of an apparently ''progressive wave'* function in a homogeneous 
medium does not insure that it represents a wave with ''progressive" physical 
characteristics. Consider, for instance, the following wave function in a homo- 
geneous medium 

(9) u = cos /3a; + OMe-'^\ 

This function may be expressed in the form (8); thus 



(10) 



A{x) = [(1.01)' - 1.02 sin' M''', ^x) = tan"^ \^^^^ ^^J- 



The phase function ^(x) is a monotonic increasing function. And yet (9) repre- 
sents primarily a standing wave. 

Thus, in general, we shall restrict the definition of the reflection coefficient 
given by (7) to conditions such as those expressed diagrammatically in Figure 
2(a), where the incident wave travels in a homogeneous medium (1) and im- 

HOMOGENEOUS INHOMOGENEOUS HOMOGENEOUS 

MEDIUM (l) MEDIUM (2) MEDIUM (3) 

-CD^IJ — ;;!: x=o V^ =^^ ^^® 

(a) 



K(x) - -^ 



x=o 

(b) 

Fig. 2. A diagram of reflection from an inhomogeneous layer (a) and a function governing 
transmission characteristics of the entire medium (b). 

pinges on an inhomogeneous layer (2) of thickness I. From x = I to x = co 
we have assumed another homogeneous medium to enable us to recognize the 
transmitted wave. Under some conditions we may be able to let I approach 
infinity. As we see from equation (7) the reflection phenomena depend on the 
properties of the function 

(11) K{x) = lf(x)/g(x)r' 
which for equation (1) becomes 

(12) K(x) = lF(x)Y'' or K(x) = lF{x)]-"\ 
according as g(x) or/(x) is independent of x. 

3. Reflection from Infinite Layers 

It is not easy to find a simple exact expression for the reflection coefficient 
from a finite inhomogeneous layer. There is no reason, of course, why we should 



120 (S184) 



ELECTROMAGNETIC WAVES 



not be content with good approximations. But there appears to be something 
in human nature, or more probably in the intellectual habits acquired in child- 
hood, that makes one yearn for the ''exact" answer to a given problem. As far 
as this writer is concerned, the exact answer is merely a specification of fairly 
easy successive steps by which one can obtain increasingly better approximations 
until further improvement loses its meaning. We except, of course, those trivial 
cases in which the exact answers are given by integers. And in practice it makes 
little difference whether a given problem is solved approximately or replaced by 
an approximating problem which is then solved exactly. 

Nevertheless in the case of wave propagation there is a class of idealized 
problems which at first sight appear to resemble closely important practical 
problems and which can be solved in closed form in terms of elementary func- 
tions. There is a temptation to apply these solutions to the practical problems, 
and we wish to warn the readers against such applications. For instance, if the 
dielectric constant is given by^ 



(13) e(x) = 61 + e\e' + ir'[(e, - e,){e' + 1) + 63], ^ = 2Tx/h, 

equations (3) may be solved in terms of hypergeometric functions and a simple 
formula may be obtained for the reflection coefficient at a; = — 00 . If x varies 
from — 00 to 00, e{x) varies from ci to €2 . If — | €2 — ci | < €3 < | €2 — Ci |, the 



-2 



° x/h 




(a) 



(b) 



Fig. 3. Types of functional characteristics of an inhomogeneous layer for which exact solutions 

of the wave equation are known. 



variation is monotonic; otherwise, e first rises or falls to a maximum or a minimum 
and then goes back to the final value (Figure 3b). If €3 = 0, 

(14) e{x) = i(€2 + 61) + iie^ - eO tanh J^ 

and the transition is antisymmetric about a; = as sho^^Ti in Figure 3a. If 
€2 = €1 , 

(15) e{x) = ci + i €3 sech' i^, 



ip. S. Epstein, Reflection of waves in an inhomogeneous absorbing medium, Proceedings of 
the National Academy of Sciences, Volume 16, 1930, pp. 627-637. 



S. A. SCHELKUNOFF (S185) 121 

and we have a symmetric ridge or a valley in the distribution of e(x). The 
parameter ^ expresses roughly the thickness of the layer, but actually the di- 
electric constant attains its boundary values €i , €2 only ata;= — ^ , x = +00 
respectively. 

In spite of the apparent resemblance between the Epstein layer (Figure 3a) 
and a finite layer of thickness h, the reflection coefficients in the two cases are 
of different orders of magnitude if the thickness of the atmospheric layer is large 
compared with the vertical wavelength, 

(16) X. = X/A, 

where A is the angle between the incident rays and the layer, as indicated on 
Figure 1. It is known^ that the reflection coefficient for the Epstein layer varies 
as exp { — 27r/i/Xz) ; we shall show that for a finite layer it varies only as (27r/i/X^)~" 
where for small A the exponent n equals unity or two (see equations 37 and 39). 
It is only at normal incidence that n may be fairly large; but it is still true that 
the orders of magnitude of the reflection coefficients for thick layers are quite 
different in the two cases. The explanation is that as X^//i — > the infinite 
regions from ( — 0° , h) and (/i, 00 ) have an increasingly pronounced effect on 
the reflection coefl&cient at x = — 00 .. In the language of the electrical engineer 
the various sections of the medium become increasingly better ''matched" to 
each other and reflections are greatly reduced. 

If /i <3C Xa; the two coefficients are substantially the same, for as h/\^ ap- 
proaches zero the reflection coefficient approaches a value depending on the 
relative difference between the initial and final values of K and not on the 
manner of transition. 

4. Reduction of the Wave Equations to a Canonical Form 

Let us transform (4) by introducing a new dimensionless independent 
variable 

(17) ^{x) = f U{x)g{x)r dx 

'J a 

and new wave functions U{x), V{x) 

(18) U{x) = [K{x)Vu{x), V{x) = [K{x)r%{x), 
where K{x) is defined by (11). Thus we obtain 

(19) dU _ _ELu dV ^ _ijj 4_ ^ 7 

where the prime denotes differentiation with respect to the phase variable ??. 



122 (S186) 



If 



ELECTROMAGNETIC WAVES 

5. The Case of Zero Reflection 



(20) K = const., 

the solutions of (19) and (4) are respectively 

U = Ae-'^ + Be'\ V = Ae''^ - Be'\ 

u = K'^\Ae-'' + Be''), v = K-''\Ae-'' - Be'') 



(21) 



Irrespective of the dependence of the phase variable ^ on x, the ratio of the 
forward progressive wave functions is K and we have no reflections. This is 
the only case in which there are no reflections. In the case of a stratified at- 
mosphere either }{x) or g{x) is independent of x) hence K{x) depends on x and 
we must have reflections. 

It has been suggested that there is a case of variable F in (1) and hence of 
variable K in which there are no reflections. We note that (1) is a special case 
of (4) in which 



(22) 



V =^ I 



. du 
dx 



fix) = 1, g{x) = F{x), K{x) = [F{x)y 



Eliminating V from (19), we find 
d'U 



(23) 



c??? 



2= -U + 



[ S{K')' _ K^l 



U. 



The bracketed expression vanishes if 

(24) m) = (P^ + Q)-\ 

where P and Q are constants of integration. In this case, 



(25) 



U = Ae-'' + 



Ae-'' + Be'' 
P^ + Q 



Thus two seemingly progressive waves appear to exist independently in the 
medium for which K is given by (24). 

However, the reflection coefficient (7) depends on the ratio u/v. The 
second wave function in the present case is 



(■-??¥«>- -"(■+??¥«>" 



(26) 



V = A(P?? -\- Q - iP)e-" - BiP^ + Q + iP)e" . 
For a ''progressive" wave moving in the positive ?? direction we have 



(27) 



gJ=[(P. + e)^ 



iP{P0 + Q)] 



S. A. SCHELKUNOFF (S187) 123 

Hence, if the medium is homogeneous in the interval ( — «» , ^o) and then in- 
homogeneous, we have 

P 

^^^^ ^ ^ ~ P + 2i{P^o + Q) ' 

The reflection coefl^icient vanishes only at ??o = — «> . 

It is instructive to consider also the complex power flow. If u is the electric 
intensit}^ and v the magnetic intensity of an electromagnetic wave, the power 
flow in the direction normal to the stratification is 

(29) W = iuv*. 

For the "progressive'^ wave in the above case we have 

(30) w^ = i (l + WTq)^^*- 

In the vicinity of ??o = —Q/P the reactive power flow is very large. At ?? = ??o 
the power flow is infinite and this point is an effective barrier to the incoming 
wave. As equation (28) indicates, the reflection is total. 

6. First Order Reflection Coefficient 

To the extent to which K' /2K is negligible compared with unity, equations 
(21) represent approximate solutions of our wave equations. In order to obtain 
the first order reflection coefficient we convert (19) into the following set of 
integral equations 



U{d) = Uo{») - f ^ [f/(p) COS (^-v) + iViv) sin (t? - ^)] dv, 



(31) 



n^) = F„(t» + £ ^1^ [V(<p) cos (^ - ,.) + iU(<f) sin (,? - v)] Afi, 



where Uo and Vq are of the same form as U and V in (21). Let us assume that 
the inhomogeneous medium extends from t? = to ?? = 6 and that f or ?? > 9 
we have 



t<? T^/Q\ -i^ 



(32) UW = e-'% V{^) = e 

If K' /2K ^ 1, then in the first approximation these expressions hold also for 
t? < 0. To obtain the next approximation we substitute (32) in the integrands 
of (31). Hence, letting a = 9, we have 



(33) 



^(^) = «-' + ^'^fiS^-"'^- 



124 (S188) 



ELECTROMAGNETIC WAVES 



For the U function the coefficient of reflection at ?? = is 

■^ d log KM _ 



(34) 



'KM 
'0 2K{<p) 



5= f ^^e-'''d<p 

Jo 



2 Jo 



2i<p 



dtp 



dip. 



It is to be noted that is the thickness of the layer in radians. If 26 « 1, then 
(35) g ^ i log {K,/K,), K, = m), K, = K{Q). 

If log K varies linearly with § (Figure 4) 



LOG K. 



LOG K 




'^=0 



log/k^ 



a.=o 



Fig. 4. Two simple transition characteristics of an inhomogeneous layer: (a) linear, 
(b) antisymmetric cubic with a continuous tangent. 



(36) 

then 
(37) 



\ogK= logZ. +|log(/f,/K,), 



1 1 /r^ /T^ \ sin _»e 
g = - log {K2/K,) — g- e , 



If log K is an antisymmetric cubic in (t> — §0) (Figure 4), and if the first 
dei:ivative is continuous at the boundaries of the layer, 



log K = log {K,K,r' + A (^ _ 1 e) log iK,/KO 



(38) 

In this case 
(39) 



K-I'J 



log {K,/KO. 



^ , ., '0 /^ sin \ _,., 

Q = ^^7^2 [—^ - COS ej e 



3 log (K,/K,) / sin 
20^ \ 



As approaches zero, (37) and (39) approach (35). As approaches in- 
finity, the reflection coefficients in these cases vary respectivel}^ as 10 and 
1/0^. If K and its first n derivatives are continuous at the boundaries of the 
layer but the (n + l)-th derivative is discontinuous, then for sufficienth^ large 
the reflection coefficient varies as l/0""^\ This is true irrespective of the 
precise functional dependence of K on i}. A ''rapid" change in the (n + l)-th 



i 



S. A. SCHELKUNOFF (S189) 125 

derivative occurs when most of the change takes place in an interval small 
compared with the radian; such a change is equivalent to a discontinuity in this 
derivative. All this may be deduced from (34) if we integrate it by parts re- 
peatedly. ^ 

If we substitute (33) in the integrands of (31), we obtain the third terms in 
the approximate series for U and F, 

(40) 

Hence, the second approximation to the reflection coefficient is 

(41) q. = 



1+x' 



where q is given by (34) and x is the value of U2{^) for ?? = 0. The sequence 
of successive approximations can be continued and q can be expressed as a ratio 
of two series; but if the first order reflection coefficient is not small compared 
with unity, it is probably better to use the method described in the following 
section. 

7. Calculation of Strong Reflections 

Reflections increase as K'/K increases. Since /(x) and g(x) are proportional 
to frequency, d is proportional to frequency; hence, K'/K is inversely propor- 
tional to frequency. For this reason the longer waves are reflected more strongly 
than the shorter waves. Reflections depend on the rate of change of K and 
on the total change in K across the layer. If the total change in K is small, the 
reflection coefficient is small even if its rate of change per radian is large. In 
the troposphere strong reflections occur near grazing incidence, where the wave- 
length (16) in the direction normal to the layer is relatively large even though 
the wavelength in the direction of propagation may be small. In such a case 
the tropospheric layers are relatively thin in the sense that the phase of the 
wave does not change much as we pass across the layer in the normal direction. 
When the electrical engineer is faced with a similar situation in the case of 
tapered transmission lines, he replaces them by a chain of lumped parameters. 
Since his special language may not be familiar to every reader of this paper, 
we shall translate his method into mathematical terms. 

Suppose that the medium described by equations (4) extends from x = 
to X = L Let 

(42) s = l/n 

be a fraction of the total interval so small that no matter where we select a 
subinterval of length s, the change in the phase variable (17) does not exceed, 



126 (S190) 



ELECTROMAGNETIC WAVES 



let us say, 7r/4. That is, s is equal to or less than one eighth of the shortest 
wavelength in the x direction. Let us integrate the first equation of the set 
(4) in the following subintervals: (0, |s), (Js, fs), (fs, fs), • • • (l — J.s, I), 
Similarly, let us integrate the second equation of the set in the interv^als (0, s), 
{s, 2s), {2s, 3s), • • • (l — s, I). Thus we obtain 

u(0) = u(is) + i I f{x)v{x) dx, v(0) = v{s) + i 1 g{x)u{x) dx, 

^0 'JO 

u{\s) = w(fs) + i \ f(x)v(x) dx, v{s) = v{2s) + i / g{x)u{x) dx, 

(43) 

w(Z — Js) = u{l) + ^ / f(x)v{x) dx, v{l — s) = v{I) + ^ / g(x)u{x) dx. 



We now approximate the integrals as follows, 

'I 



(44) 



u{0) = uiis) + iv{0) / /(x) dx, y(0) = ?;(s) + Mis) g(x) dx, 

Jo '^ 

«f8 r.23 

w(Js) = -wds) 4- w(s) / /(x) (ia:, v(s) = z;(2s) + i!/(fs) 1 ^(x) c?a:, 

u{l - \s) = w(Z) + ivij) \ fix) dx, 

v{l — s) = v{T) + iu{l — \s) \ g{x) dx. 

J l-s 

The ratio u{0)/v{0) may be expressed as a continued fraction 



(45) 



where 



v{0) -^'-^ Y,-{--^ 1 



z,+ 



¥2+ '" + 



1 



7 , !^' 



(46) 



Z^ = i fix) dx, Z2 = i fix) dx, • • • , Z„+i = i / /(: 

/»8 /»2« 

Yi = i gix) dx, ¥2 = 1 gix) 

^0 ''a 



> > 



Yr, = i gix) dx. 

h-3 



S. A. SCHELKUNOFF (S191) 127 

Knowing this ratio and K, = [/(0)/^(0)]'/' and u{l)Ml) = [f{l)/g{l)Y'', we 
find the reflection coefficient from (7). 

In the case of electromagnetic waves given by (3) we find it convenient to 
normalize the equations by setting 

(47) E = u, A-%/e,y''H, = v, 
so that 

(48) fix) = 2;r/X3 , g(x) = {2t/K)[i + ^^"^ ~. "'^ ], 

where X;, is given by (16). If we choose s == IK, then 

Zi = Zn+i = itt/S, Z2 = Zs = " ' = Zn = ^V/4, 

(49) 



''■=sC['+*^"=-]'^' 



If for X > I, the medium is inhomogeneous but such that K'/2K is small, 
then u(l)/v{l) does not quite equal 1/(0/^(0]^''^. In terms of the reflection co- 
efficient 

we then find 

(51) ^ = f^[/a)/?a)r. 

Equation (45) is needed only for that part of the inhomogeneous medium for 
which K'{x)/K{x) is not small compared with unity. 

8. The Differential Equation for the Reflection Coefficient 

The equivalence between linear differential equations of the second order 
(or systems of two linear equations of the first order) and the generalized Riccati 
equation is well known but the physical significance of this equivalence is not. 
While the wave functions satisfy the former equations, the wave impedance and 
the reflection coefficient satisfy Riccati equations.^ Thus equation (7) defines 
the reflection coefficient dX x = Xx in terms of the ratio (the "wave impedance") 
Zi(xi) = u{xi)/v(xi) and K = [fixi)/g{xi)Y^^. From (4) we find for any x 

,-^. dZi 1 du u dv '4-/ \ i ■ r w^ 



3J. R. Pierce, A note on the transmission line equation in terms of impedance, Bell System 
Technical Journal, Volume 22, 1943, pp. 263-265; in this paper equation (52) is obtained di- 
rectly from physical considerations. L. R. Walker and N. Wax, Non-uniform transmission lines 
and reflection coefficients, Journal of Applied Physics, Volume 17, 1946, pp. 1043-1045; in this 
paper equation (53) is derived and several applications are given. 



128 (S192) 



ELECTROMAGNETIC WAVES 



This is the differential equation for Zi{x). From (7) and (52) we obtain 



(53) 



dq 



f^ = 2im^--{i-<f) 



dlogK 
dx 



If the layer extends from a; = o^i to a; = 0^2 , then q{x2) = 0. Together with this 
initial condition, equation (53) determines the reflection coefficient q{xi). 

It is more convenient to express the above equations in terms of the distance 
from the upper boundary of the layer. This is equivalent to reversing the sign 
of X in the foregoing equations, 



dZj 
dx 



dq 
dx 



(54) ^ = ij{x) - ig{x)Z\ , ^ = -2imq 
Introducing the phase variable ^{x), we have 



(1 



2, dlogg 
^' dx 



(55) f = ,(K-a l!=-2.-,-i(l 



K^ 



d^ 



q') 



dlogK 
d^ 



If I g I <3C 1, we may neglect q^ and obtain an approximation for q equivalent 
to (34). We can then obtain higher order approximations by the perturbation 
method. Alternatively, we can easily obtain q numerically or on a differential 
analyzer. 



The Theory of Magneto Ionic Triple Splitting 

By 0. E. H. RYDBECK 

Research Laboratory of Electronics, 

Chalmers University of Technology, 

Gothenburg, Sweden 

1. Intraduction and Summary 

Triple splitting of ionospheric waves or rays was reported in 1933 by 
Eckersley [1]. A few years later, in 1935 and 1936, Toshniwal [2] and Harang 
[3] reported similar phenomena. However, no serious attempt was made to 
explain them theoretically. This interesting matter did not attract much 
attention until after world war II, when the chain of Recording Ionospheric 
Observatories had been greatly increased. 

According to Meek [4] triple splitting was observed in Canada (1943) when 
the Churchill station commenced operation. Since then the phenomenon was 
observed at other stations as well. In Hobart, Tasmania, Newstead (1946) 
observed Fz triple splitting [5]. Seaton, reporting from College, Alaska in 1947 
presents a record of F2 triple splitting [6] and discusses the possible mechanism 
of excitation of the third or ^-component of the down-coming wave. 

At the Kiruna Ionospheric and Radio Wave Propagation Observatory 
(67° 50' N, 20° 14. 5' E), which started regular recording on October 1st 1948, 
a large number of E- and F-layer triple splits have been recorded [7]. As far 
as is known to us, E-triple splitting has only been reported earlier by Meek [4] 
who is also working at high latitudes. The frequent occurrence of ionospheric 
triple splitting at Kiruna motivated us to attempt a theoretical explanation of 
the phenomenon. The first outline of the theory, based on the coupled 0- and 
x-wave equations of the author's treatise, "On the Propagation of Radio Waves^\ 
(Gothenburg 1944), was presented at the annual meeting of the Indian Academy 
of Sciences in Bombay, December 1949 and was subsequently published [8]. 
The further theory, including a description of the 2;-ray paths, was presented 
at a session of the Societe de Radioelectriciens in Paris, February 1950. 

This theory, which forms the main contents of the following conmiunica- 
tion, proves that for any ionosphere, even a smooth one, a 2-wave is excited by 
the vertically incident ordinary wave in the neighbourhood of and at the ordinary 
reflection level, at moderately high geomagnetic latitudes. At sufficiently high 
geomagnetic latitudes this coupling becomes very strong. One can then, for 

Paper presented at the June, 1950, Symposium on the Theory of Electromagnetic Waves, under 
the sponsorship of the Washington Square College of Arts and Science and the Institute for 
Mathematics and Mechanics of New York University and the Geophysical Research Directorate 
of the Air Force Cambridge Research Laboratories. 

129 (S193) 



130 (S194) ELECTROMAGNETIC WAVES 

vertical incidence, picture the ionized layer as consisting of coupled o- and 
2-transmission lines. Finally, at a critical geomagnetic latitude, for which the 
inclination of the terrestrial magnetic field, ^i , is given by 

e. = tan-' ii([l + {<^H/vfr - l)!"^ 

w^ere v is the electronic coUisional frequency and coh the angular gjTofrequency 
of the free electrons in the terrestrial magnetic field, the entire o-wave is trans- 
formed into a z-wave and the ordinary trace on the ionospheric record becomes 
weak. For latitudes higher than the critical one, which for v = 2.10'' c/s is about 
67°, the o-wave rapidly disappears. Only the z-and x-components remain and 
the propagation is purely longitudinal. This is in very good agreement vrith 
the experimental results. 

It is further shown that at low geomagnetic latitudes, where the normal 
coupling would be far too weak to explain the rare cases of low latitude triple 
splits, a strong step-like increase in the electron density can produce a 2- wave 
of recordable strength, provided this step occurs practically at the regular 
interaction level and has an extension of about one vacuum wave length. Ir- 
regularities in the electron density distribution may therefore, under rare cir- 
cumstances, be effective in producing triple splits at low latitudes. 

2. General Consideration 

To begin with we assume that the terrestrial magnetic field of strength H 
lies in the z,y-p\sine of the rectangular coordinate system yviih components H • cos 6 
and H-sm 6 along the respective axes. If z were chosen to be the vertical, 
7r/2 — 6 then would denote the inclination of the terrestrial magnetic field, ^, . 

Introducing the average collisional frequency, v, of the electrons, co^ = 
their angular gyrofrequency in the terrestrial magnetic field and co^ = the critical 
angular frequency of the ionized medium (of specific density N cm~^ of the 
free electrons) in the absence of the magnetic field, we can ^Tite the familiar 
relations between the x,y,z-covciiponents of the electric field strength E, and the 
polarization, P, in the following form, viz. [9, p. 8] (time factor e~'"'') 

(1) -EJ^TT = x\l + J8)-P^ - jjr'P. + JJL'P, 

-EJ^T = x\l +J8)'P, - jyj^'P^ 
where x"^ = wV^c, Tr = x'^-y-sm 6, jl = x^-y-cos 6, 8 = v/co 

and y = cch/o). 

We next assume that z is the vertical axis and that the electron density, 
A^, is a function of z (height) only. If we further assume that the incidence of 



O. E. H. RYDBECK (S195) 131 

the (plane) electromagnetic wave is vertical the wave normal will also be vertical, 
although the direction of maximum energy transport may be more or less oblique 
(see also Section 6). We therefore put d/dy = = d/dx. The condition div 
D = 0, where D = E + 47rP is the displacement vector, thus means that E^ = 
— 47rF, , as we do not consider constant fields. Under these circumstances we 
find 

-EJ^T = {a + 7?/(l - «)} -P. + jyL'Py , 

(la) -^,/47r = a-P, -jVl-Px, 

where 

a = x\l +j8). 

Now for a certain value, u, of the polarization ratio Py/Px , the ratios 
Ex/Px and E^jP^ become identical and consequently must belong to the same 
wave solution. This is very important because it is in a sense the foundation 
of the following analysis. One immediately finds that two i^-values, u^ and u^ , 
are possible, viz. 

(2) ^; = 37^+7il«»T(«^ + (-^'+i«)')"'!. «i-"2 = l 

where F^ = Xjx — 1 and 5^ = vji^. Here v^ = co-?/-(sin^ 0)/(2-cos 6) is the 
critical collisional frequency of Appleton-Builder [10]. 

As V^ is a function of z (as are also dc and 5) the w-ratios are not independent 
of height. This means, as we will soon see, that the two wave-solutions corre- 
sponding to Ui and U2 generally are not independent of each other. Since 
EJP, = EJP, , we have 

(3) El''/E'x'' = u, , El''/E':' =u, = l/u, . 

When coe = cu, F' = 0, and u^ = [vjv - {{vjvf - iy^% From this relation 
one recognizes the old fact that when co^ = co the polarization angle 4> (tan 
is equal to the ratio between minor and major axes) is zero when v < v^ (plane 
polarization) and db tan "^ [{v — v^jiy -\- v^'^''^ when v > v^ * 
From Maxwell's equations we find [9, p. 12] 

2 r) , ^^ _ Q 

where /cq = 27r/Xo and Xq is the vacuum wave-length. 

For the two w- values we further easily find from (1) that 

(5) £>:" = ^,-E['\ Dr = .,-ET, Dl" = ,rEl'\ D[" = .,-ET , 

where €1 and ca are the complex dielectric constants of Applet on-Hartree for 



132 (S196) ELECTROMAGNETIC WAVES 

the ordinary and the extraordinary waves. In our notations it is very con- 
venient to write €i and €2 in the following forms, viz. 

'' ^ "' ^ 1 {l+.iS-B)B 



, , '■ il+j&)B-ycos0lS.^{dl+BT'] 

€2 /t'2 

(6) 

(1 + J8)B - 2/- cos e[8, =F (5! + B^'Y 

where -B = —V^+jd and index (1) denotes the ordinary component, etc. 

As D, = D^'^ + Z)f ' and D, = Dl'^ + Z)i'\ one finds that the formaUy 
simple wave equations (4) yield the two coupled wave equations 



(7) 


'^^^ _L C7.2 /2MT TT '^^ ^^2 f 


where 




(7a) 


n, = £i"/(l - M?)"^ and n^ = El"/{1 - ul) 


One further has (with the chosen time-factor) 


(8) 


'^ 2 dz {V - ja,){V' + ja,y 


with 




(9) 


Gi = S, + S, 
a2 = Se — S. 



mations^ 



3. The Unperturbed Wave Functions and the 
Coupling Coefficient 

For the unperturbed o-wave functions we use the very accurate approxi- 

(10) 

^Generally very accurate except in the neighborhood of the second zero of ei . When two 
zeros come very close (not of immediate interest in this connection) parabolic type wave 
functions have to be used, see reference [11], p. 23. 



O. E. H. RYDBECK (S197) 133^ 

where Wi is the phase integral 

(11) W, = r ko-n,-dz= W,{z^,z), 

and Za, the location of the first zero of ci . 

Let us assume that the ionized layer is formed in the height interval 
{zq — A/i) — (2:0 + A/i), where Zq denotes the centre of the layer. If further 
ni = e~" at Zo — A/i, then nj'" represents a wave incident upon the layer 
from the —z side {z assumed positive in direction upwards). Neglecting the 
possible formation of a 2- wave the coefficient of reflection at the lower boundary 
Zq — Ah asymptotically becomes 

(12) R - exp {j'2[W,{z^ , z, - Ah) - 7r/4]}, 

if the layer is not very thin (counted in Xo) and co 5^ co^ . 

For the unperturbed s-wave functions we use similar wave functions denoted 

by UV'\ Ul''' with 

(13) TF2 = f ko-n^'dz = W^iz, ,z), 

where Zf, is the position of the appropriate zero. It appears that it is of funda- 
mental importance to study the behaviour of Ui and ng in the 2-plane. 

We denote by Za„ the interaction level where oj^ = w (also branch point of 
Til Avhen V = 0) and use the following approximation in the neighbourhood of 
this level, viz. 

Under these circimistances we have 

(15) xjy = -i ^-^^ 



2 {z - Za, - ja,/jaA{z - Za, + ja2HaoV 

because dV^/dz varies only slightly through the narrow interaction region. It 
is extremely interesting from an interaction point of view to find that 

^ + co I ^' V > Vc 

(16) rPdz={ 

(-jV/2, v<v,. 

As n\ is zero for V^ = jb we immediately find that the branch point Za of 
Til is mid-way between the two poles Zi = Za, + jcii/yao and Z2 = Za, — jO'2/ya^ 
of yp, i.e. 

(17) 6m (Za) = Za- Za, = (flj " a^l2ya. = 8/ya, . 



134 (S198) ELECTROMAGNETIC WAVES 

It is further important to note that 

1) fe).=.. = 1, 



and 



2) 



712 at the poles. 



The two zeros Zi, and z^ of nl are found for V^ = y + j8 and V^ = —y + j8. 
They are thus independent of d. 

If we assume a parabohc ionized layer of the form 



coc =o;L{l - [{z -Zo)/AhY 



where co^^ is the angular critical frequency of the layer (for the moment assimied 
larger than w (1 + y)), we have ^m{z,) < dm(Za) < ^m(zi,) as depicted in Figure 
1, where the poles of xp are also shown. 



z -plane 



^c 



---^^ 






'a 



\ 



'^■-*\ 



'b. 



^o 






I 

Fig. 1. Location of the zeros of n^ and the poles of ^ in the z-plane. 

Finally it should be noted that | ?i2 1 = oo for 2 = 2^ , at which point (branch 
point of Rai) 






(18) 



(cOc/co)^=,, = 



1 +i5 



1 + 



sm" Q 



(1 + ;•«)' 



In Figure 2 we have plotted the variation of €i and €2 for i^ = 0, a normal 
electron density gradient and geomagnetic conditions of the Kiruna Observatory'. 
The unperturbed standing wave functions 



nr" =nr^^+nr^' 



and n^" = n^^^ + n°^'', 



O. E. H. RYDBECK 



(S199) 135 



corresponding to these €-variations are shown in Figure 3 ior v = 0, together 
with —^m (xl/) for v = 0.95 -j/c . This figure demonstrates that the couphng 
actually takes place within a very narrow region of a few hundred metres 
effective extension. 




sin^e 



f m J. 47 Mc/sec 

f^ - /.-^^ Mc/sec 
e. '77*3' 



'103 m 



Fig. 2. Index of refraction in the neighbourhood of ojc 



when 



0. 



y^^=-S.3Al0^c/sec 
a, m 2V-3A7 Mc/see 

Aq. as.sm 

77'j' 

i.'^A Mc/sec 




Fig. 3. The coupling coefficient and the unperturbed wave functions in the interaction region. 



i 



136 (S200) 



ELECTROMAGNETIC WAVES 



It is of particular interest in this connection to demonstrate the pecuHar 
nature of yp for various i^-values. The variation of the important component, 
—^m (i/'), is therefore shown in Figure 4 from which it is easy to understand 
that the ''interaction integral" (16) is zero when v > v^ » 



-Im(f) 



\J^'2.3A /O^c/sec 
CO •2ir-3.A7 Mc/sec 



-^ 




e, = 77 3 

Unperturbed ordinary standing wave (\f*6) 



*700 



■750 
\ 



•1000 z-z„ meters 



Fig. 4. Demonstrating the nature of —^m{\(/). 

4. First and Higher Order Approximations of the 
Z-Wave (Third Component) 

In order to study the first order approximation of the excited ;s-wave we 
introduce 



(19) 



rOCO 



o('> 



^V\z) = -nT 'dyp/dz - 2-il^-dUV /dz, etc. 



It is necessary to treat the pair nj^'\ 11?^'^ together, as only the sum of these 
two wave functions will yield an exponentially decreasing tail along the positive 



J 



O. E. H. RYDBECK (S201) 137 

real axis {z real and >ZaJ. The first order 2-wave propagating in positive 
direction thus becomes 

(20) nr" = ~nr--, (' nV'-iil" + ii") dz, 

where wl is the Wronskian of the pair Ila^'', Ul^'^ With our form of functions 
wl — j' 2kQ . Similarly for the wave running in opposite direction we obtain 

(21) nr = nr-\ r nr "•(?;■' + {i^v^. 

W2 '^b. 

In these formulae 61 and 62 denote lower and upper boundaries of the effective 
interaction region. 

It is convenient to write these integrals the following way, viz. 

W2 ''bt 

(22) H- nr"-nr"-(nr"Vnr" - nr^/uD^} dz 

W2 ''bi 

and similarly 

U2 =112 •-;• {ni 'U2 -(Hi yui - U2 yu2 )yp 

W2 ''b, 

(23) 

+ uT''ii'2''''(iiT'''/ur - ur'/uV')4^} dz. 

For the moment we are principally interested in equation (22) which yields 
the first order approximation of the wave propagating towards Zi, . As far as 
the interaction integral is concerned, replacing 61 and 62 by — <» and + 00 should 
not appreciably affect its value if z does not lie between 6x and 62 and these 
boundaries are well outside the main interaction zone. 

If we now deform the path of integration to follow mainly the straight line 
^m (V^) = —j(a2 -\- e) = — j{(5<. — 8) + e}, where e is supposed to be a very 
small quantity, it is possible to show that the main contribution to the inter- 
action integral comes from the residue at the pole —ja2 of \^. As a matter of 
fact, the remaining contribution (from the rest of the path) is of the order of 
the second term in the asymptotic expansion of the dominant wave function 
at the pole. When therefore ^ > i^^ , H^'' = 0. 

In spite of the fact that the poles of xf/ are also branch points of Ui and 712 
(where the cuts joining their Riemann surfaces begin) we can use our wave 
function approximations with good accuracy at these points when the coupling 
is small (\\f/ \ small along axis of reals). This can be demonstrated approxi- 



138 (S202) ELECTROMAGNETIC WAVES 

mately in the following manner. Introducing Ui = [I -\- li{z)Y^^, we know 
that El' , n/ are exact solutions of the wave equation 

di'U/dz^ + (kl-nl - Q)U = 0, 

where 

Q = -r'iz)/Ml + l(z)] + 5/m[nz)/[l + l(z)]Y - 5/4(TF(/3TF,)^ 

As (8l + BY^^ = -[{Vy'8j2M'^^ (appearing in l{z); see equation (6)), one 
finds that Q contains terms of type xj/^^^, \p, xf/^^^. To add these terms to the original 
wave equations should hardly affect the result when xj/ is small. The approxima- 
tion is slightly rougher than the neglect of the xp^ term actually present in the 
wave equations. 

The distance between the poles and z^ is quite large. For the locality in 
question (Kiruna) Vc = 2.1-10^ sec~\ and with the electron density gradient 
chosen this distance is greater than 100 metres. The amplitude of TTi at the 
poles therefore will be of the order 3. The corresponding phase angle, with the 
chosen orientation of the n-planes, Hes between — 7r/2 and — 37r/4 at z^ and 
between 7r/2 and 37r/4 at Z2 . For TF2 the amplitude is several times as large 
with phase angles approximately ^7r/2. 

It is therefore possible to use the asjnnptotic representations of the Hankel- 
functions at the poles. As rii = na at the poles, one infers that the first term 
of (22) practically disappears. Virtually only the incident wave function pro- 
duces a 2;-wave into the layer. It should be borne in mind, namely, that the 
reflected wave function, nj ' must be present to make the integral practically 
zero over the rest of the path. Furthermore, in a region of rapidly changing 
refractive index, only a special treatment of IlJ ' and IlJ ' \\ill disclose how 
much of them is to be regarded as momentary waves in both directions [11]. 

At the poles 

[III /III — II2 /II2 \ ^ 2jko-nj,oie , 
and therefore 

(24) Ul''\z) ^ U',''\z)-exp {-j[W,{z^ , z,) - W,{z, , z.^)]} ■e-'^''-T/2, z > h„ 

We should have started with unit amplitude of the incident wave at Zq — Ah 
(the layer boundary). The nj''\2;) -function of the interaction integral should 
therefore be replaced by uT^\z)/uT'\zo -Ah). If > 62 , but z 7^ z, this 
yields the useful result 

Ul''\z) ^ (^-Y'' exp {j[W,{z.^ , zo - Ah) + Tr2(^, zj]} 
(24a) 

• I exp jil TF2(2a„ , Z2) - W^{z,, , z-^ - Ijj. 

As the first exponential is the straight phase integral up to z (though with 



O. E. H. RYDBECK (S203) 139 

exchanged refractive indices after the 2- wave 'Hakes over") we are led to intro- 
duce the first order transmission coefficient 7"+ ^ of direction into the layer, viz. 



(25) 



rr ^ ^ exp ^ j\ Wlz^^ , z,) - TF.fc, , z,) - |] ^ v < v. 



n =0, V > V, . 

As Z2 — Za^ ^ —j{vr. — ^)/co-7a„ , We scc that I TV ^ I approaches x/2 as p 
approaches v^ . This indicates a very strong, coupling as indeed one would 
expect. When v approaches Vc the mathematical reflection point, Za , comes 
farther and farther away from the physical interaction point z^^ . At the same 
time the interaction pole, Z2 , comes closer and closer to the physical interaction 
point and more and more of the incident wave energy is transformed into a 
2- wave. Finally, at the point of ''resonance" (v = Vc) there appears to be 
practically complete transformation. When the coupling becomes so strong, 
however, our first order approximations cannot be accurate enough and it is 
surprising that they are as good as they are. In (25) one would only expect a 
coefficient of about 1 instead of 7r/2 when v approaches Vc . However, it is not 
too complicated to obtain approximate higher order approximations if we make 
use of the fact that 

(26) n;'" = ^- f fMHs) ds-l [' mm dr, 



C02-C01 ''bx 
r(l) 

mate total transmission coefficient 



etc., where III ' denotes the first order correction in nj 

Introducing T+^ = 7r/2-exp { — j7r/2 — r), we find the following approxi- 



rp r^T^ -ir/2 f ^ 

(27) ^ ^ 16 ^ 



n ^ 0, v> V, . 

For V = V, this yields (rO.=.c = 7r/2/(l + ttVIG) ^ 0.97. For | e'^ = 1/2 
(the imaginary part of r is quite small as will soon be shown) similarly \T+\ = 
0.72. 

As Ui and ^2 are representations of the same function in different, inter- 
connected Riemann-surfaces only (these properties have been discussed in an 
interesting recent paper by T. L. Eckersley [12] it might be of interest to search 
for a connection formula between the wave functions in the principal Ui and 
n2-planes. A complete connection would yield as result a formula containing 
all waves running up and down and between the various branch points. Such 
general connections, which are very difficult, are not of immediate technical 
interest as many of the components are highly attenuated and it is therefore 
inadequate to try to connect the waves reflected at z^ and z^ . 



140 (S204) 



ELECTROMAGNETIC WAVES 



If the 2-wave functions n 



0(0 



nr 



are "carried" through the cut round 



2a back to the axis of reals, but now in the o-plane, one immediately infers from 
the asjrmptotic expansions of these functions that they transfer to o- waves 



vai 



z- plane 



o- section 



z-secfion 



p, 



'Oo I 



% 



r- 



Real axis 



G^ 



Fig. 5. Transition cuts between two of the Riemann-surfaces. 

along the axis of reals at a sufficient distance from z^^ . The transformation 
coefficient apparently gets the ''geometrical" value 

exp {TjW^feo , z,) - Wlz.^ , z,)]} = e^\ 

Applying the methods of connection already used by the present author 
[11] we connect the two function groups IlJ ' + A-II? ' and B(n.y +112 )• 
The effective reflection coefficient then becomes Reff = A-R, where R is the 
regular reflection coefficient (12) when the 2- waves are not considered. One 
finds when v < Vr that 



(28) /?eff ^ exp {2jlW,{z^ , z) - 7r/4]Kl + u/2)/{l - u^l2), v < v. 



O. E. H. RYDBECK 



(S205) 141 



where 



u' = e-'^ exp {2jlW,{z, , zj - W,{z^ , zj]}. 



As I w^ I <C 1 we have approxhnately 



R.ii^ 



exp l2j[W,{z. , z) -7r/4]t 



(28a) 



+ 



Rq 



e-'^-exp l2j[W,(z, ,zj + W,(z^^ , z)]\ 



Z' plane 




^^C-^ real axis 



Fig. 6. Depicting complex of waves contained in the effective reflection coefficient. 



Expansion (28) shows that it is possible to consider the branch-points as 
having local ''geometrical" reflection coefficients e*''^^ and transmission co- 
efficients 2^''^-e"^''^*. The ''geometrical" attenuation from branch-point to 
branch-point becomes d = \w2 -\r \^m [W^iz^^ , z^ — Wx{z^^ , ^2)) |- It must 
be borne in mind that this representation is a mere construction. However, it 
is possible to give these "constructed" coefficients a simple physical interpreta- 
tion [11, p. 17]. 

When V > Vc 1 the real axis passes straight through the cut into the bi- 
section and the o-wave automatically is "deformed" into a 2- wave, i.e. we have 



142 (S206) 



ELECTROMAGNETIC WAVES 



longitudinal transmission. This simply explains why T+ = 0, when v > v^ 
The connection relations now yield 

u" = exp {2jWi{Zb ,z)}, V > V, 

and thus approximately 

roQ^ p - ^^P !2jWife , z) - V4]! , exp {2j[W,{z, , z) - 7r/4]} 



Re 



R. 



Of these terms the first one represents that which still remains reflected from 
around the 0-level of reflection. This term disappears fairly rapidly but still 



lm<f> 




Fig. 7. Depicting the variation of ni and n^ through the coupHng range. 



exists when v > v^ . The triple split therefore does not abruptly change to a 
double split type z, x when v > v^ . This is also clearly sho^Mi in Figures 7, 8 
and 9 which depict the real and imaginary components of ni and ??2 as functions 
of 2 — Za^ along the real axis through the coupling range. 

When V = Vc the transition from the n^- to the ??2-branch is clearly shown. 
This transition makes dn^jdz = oo = dn2/dz when z = Za^ . This produces a 
considerable part of the 0-level reflection still present when v = Vc . ^Mien 
V = 1.05 -^c , Figure 9 shows that there remains a fairly noticeable change in 
nx around the Zo„-level. This produces the reflection mathematically represented 
by the first term in (29) . 

Finally, from (27), (28) and (29) it appears that with respect to the com- 



O. E. H. RYDBECK 



(S207) 143 



\f^m 2,2/ • /O^ c/see 
f m 3:75 Mc /see 




Fig. 8. Depicting the variation of n\ and rii through the coupHng range. 




Fig. 9. Depicting the variation of n\ and n% through the coupUng range. 



144 (S208) 



ELECTROMAGNETIC WAVES 



ponent reflected from the 2-level {i.e. at Zj,) the effective transmission coefficient 
becomes 



(30) 



Toz^ 



2^ 



e ', 



1 -1- — -e"'" 
^ ^ 16 ^ 



.1, 



(one finds r_ = -T^) 

(v small) j 
(p large); 



For all practical applications it is sufficient to use the simple form 



The fact that the coupling between the wave function disappears when 
V > Vc \s natural, as has already been stated, because we have longitudinal 




Fig. 10. Depicting the variation of the polarization ajigle through the coupling region. 

transmission when v > v^ , However, if we take a look at Figure 10 depicting 
the variation of the polarization angles through the coupling region, we vdW 
find another way of explaiaing the phenomenon. Only when v < v^ do the 



O. E. H. RYDBECK 



(S209) 145 



polarization curves cross each other, which appears at z^^ . It is easy to under- 
stand that coupling is much more likely to take place between the Avaves when 
the polarization is the same than when it is widely different. Even if coupling 
were present, one infers from the shape of the ^a-curves when v > Vc (Figure 9) 
that no wave of any significance could be excited or propagate. 

5. The Transmission Coefficient as a Function of v 

In accordance with (30) we have 



(30a) 



To' ^ exp < j 



b" C 



{712 — ni) dz =b 7r/2 



V '^ Vc 



Introducing — T^ + jb = jixbc , we obtain the following simple expressions 
(31) 



n\ 



nl 



1 T (1 - ^ y + (l/2)M'-tan^ 6 
1=F(1 -MT''->(H-i5)/cos0' 



where m varies from —1 to +1, when z varies from z to Z2 (along the imaginary 
axis). In Figure 11 we have plotted the variation of rii and ^2 along the pole 





^^ 


^__^,..,^ «±>/ 




-^ 


^ 




^ 


t~3.47Mclstc 
1^ - /,'>•* Mcjsee 










"-"^'/N 


"^ 


«. '77'3- 


Ct'0,5 


.-^^ 




^ 


Re(n) ^r 


"/' 


* 


>, 


^y- 






"2> 


3 


'0.5 








*0.5 


^ *'.0 


-— ' 















Fig. 11. Showing the variation of refractive indices along the pole axis. 

axis. We note the small difference between ^m (na) and ^m (ni)^ which indicates 
that the real part of the phase integral in (30a) is very small. 



146 (S210) ELECTROMAGNETIC WAVES 

With fi = v/vc we can write 
(30b) To' ^ exp {jlkodXl - y)M{n)/y^^ ± 7r/2] - kM^ - ^J)N{JJ)h^^, 



where 

(32) 

and 

(33) 



1 C 

Mill) = —~^ — / [^m (ns) — ^m (ui)} dix, 

1 r' 

N(ij) = :j-3— {(Re (ris) - (Re (nj) (?/z. 



In accordance with Figure 12, which depicts the variation of the mean values 
of refractive index differences (for the Kiruna values used throughout) M{fi) 
can be safely neglected, being of the order of 0.05 only. 













'i>/,0 


N(/J> 












1 -J. 47 A*/sec 








<>v " '^^ f^^l^^ 








q.'Tt'j' 


<±> OS 




-^.^ 


'/s 

N(o) (i-y> 






N(/J) / 


^^~\\ 

^\\ 






M(/j) 


^\\. . ^'/^> 






■— — " ~"""-^N?^ 


'±'0 05 








^^^ 










M ^ 





Fig. 12. Depicting the mean values of refractive index differences as functions of ^ along 

the pole axis. 

Therefore with good accuracy 
(31a) Tl'T-,, ^ exp { -2kMl - n)N(n)/y^^. 

From (31a) we finally obtain for the parabolic layer (of half thickness Ah) 
the practically useful formula 

(34) n^To. ^ exp {-vMil - vMN(vM(Wo^,J/UJc/ - 1^'], co < co,. 



O. E. H. RYDBECK 



(S211) 147 



where Co is the velocity of light in vacuum. At least for Kiruna conditions 
Niij) according to Figure 12 is well represented by iV(0)(l — nY^^ for practical 
purposes.^ We therefore obtain the very useful practical approximation of (34), 
viz. 

(35) I To. r = exp i-..A/.(l - .AJ^/^-7V(0)(o./c..J/[coL/co^ - if']- 
It is apparent from Figure 12, relations (34) and (35), that 

{d I To. 17^^).=.. = 0. 

Figure 13 illustrates the result (34) for typical £'-layer values and Kiruna geo- 
magnetic conditions. 























IrJ ■ 




















1 1 


^ 








.,:' 


^-^>^ 












X 










/o"^ 
























/^ 




Ah '•JOkm 

It - 3.3 lO^/sec 






-J 










f =3.A7 Mcjsec 
























-4 

10 














^K 





Fig. 13. Total traixsmission coefficient as a function of v. 



In Figure 14 we have plotted the approximate variation of v^ with geo- 
magnetic latitude. For comparison the ionospheric height corresponding to 
the i/-scale has also been included. First, when we come to a geomagnetic 
latitude of say 60° (the approximate geomagnetic latitude of Oslo) does the 
longitudinal propagation develop in the lower £^-layer for vertical incidence. 
For low, high and very high geomagnetic latitudes the ionospheric virtual height 



2The possibilities of approximating A^(/x, Q) by A^(0, ^)(1 — /i)^^^ over a wider range of Q 
values will be investigated shortly. 



148 (S212) 



ELECTROMAGNETIC WAVES 



traces, as functions of wave frequency, will therefore, in accordance with the 
theory presented, appear as shown in Figure 15. This is in good agreement 
with experimental results. 





Critical 

collisiona 
frequencL) 


/ 












1 r 

Do tj light values 1 


^c 










^ / - 
















1 / 
















i / 
















1 / 
















/ 
















/ 












1 


lO' 












! / 


















/ : 


















/ 1 
















1 / 1 
















1 /> 
















1 / I ; 
















1 / 1 


















/ 


















/ ' 


10^ 
















/ ' 1 














/ 


/ 


















y 




. 














^ 


k 


















^ 




















^ 




















^ 


















^ 






1 


















/o'> 


























^ 








■ 








^^-"^ 






























■ 






^ 


















/ 
















y 


















/ 


































JO' 


















/ 














1 r 



220 



Height 
km 



120 



lOQ- 



80 



Geomagnetic latitude 



Figure 14 



At the Kiruna Observatory, where v^ = 2.1-10^ c/s, triple splitting is 
observed very frequently in the central and upper regions of the ^-layer. It is 
observed regularly in the early mornings and late afternoons when the ionosphere 
is quiet or only moderately disturbed. The frequency of F^ triple spHts is 
smaller. At lower geomagnetic latitudes (southern Scandinavia) the situation 
generally is reversed. Therefore, let us for a moment study the ratio Rz/Rq . 
Assuming that for v < v^ 



^m [W2{z, , 2, J} - ^m [W,{z, , zj} \«\^m [W,{z,, , zj] \ 



we have 



O. E. H. RYDBECK 

RJRo I ^ exp <j -2ko\ \ ^m (na) dz 



(36) 



+ 6,(1 - iiY^''N{0)h 



••]}= 



(S213) 149 



V < Vr 



Further assuming that (as far as order of magnitude is concerned) for higher 
latitudes 

/ 6m {rii) dz - {Zi>, - Za,) 6/ci/(l + y), 
where ki may be smaller or larger than one, and noting that at least approxi- 



2 O X 




Z O X 




Too weak to 
be recorded 




High latitudes but v^^ > v» 



very high latitudes Cw'^ < ^> 



Fig. 15. Depicting ionospheric virtual heights as functions of wave frequency for 
low, high and very high geomagnetic latitudes. 

mately {z^,, — Za,)d{o3l/w)/dz ^ ica-?/, where /C2 probably may be somewhat larger 
than one, we find 



150 (S214) ELECTROMAGNETIC WAVES 

X 

A normal value of A^(0) is about 0.5 (for the Kiruna conditions chosen actually 
0.6, see Figure 12). For ^-layer conditions /Ci-/C22//(1 + y) may be about the 
same and for the i^-layer considerably smaller. This means that for the E- 
IsLjev X may be only slightly varying whereas it should decrease ^dth increasing 
ju, v/vc , for the F-region. This means that in the F-region one is likely to record 
low level triple splits more frequently than at high levels. This is in substantial 
agreement with our Kiruna results. 

A comparison between the E- and F-layers depends very much upon the 
respective gradients Too . At Kiruna ja^ often is so large for the auroral £'-layer 
that q becomes quite small. At lower latitudes this is not so and as x moreover 
is smaller for the i^-layer, i^-triple splits may be recorded more frequently. 
This is in good agreement with results in southern Scandinavia. In all cases, 
places with low Vc (see relation (37)), i.e. at very high latitudes, are the places 
where strong triple splits are likely to occur. However, triple splits have been 
reported from places where Vc is so large as to make q too great for any triple 
splits to be recorded. One therefore naturally asks the question if a sudden 
change in electron density, almost a discontinuity, might not produce the de- 
sired — z coupling at the lower latitudes. A brief investigation will show that 
this is not generally so. 

Let us assume that V^ has the form 

(38) V^ = jao'Z + /3-tanh {{z — e)/xo], (3, e, and Xq real. 

The smaller Xq becomes, the more step-like becomes the electron density dis- 
tribution. If (e/xoy <<C 1, the step function can be considered as lying within 
the main interaction range. If, on the other hand, (e/xoY ^ 1 the step Hes 
outside the interaction range. Further, it should be noted that the poles of 
V^, z = e + jxoiTr/2 zt n-ir), are not poles of xf/. Therefore, as far as the pro- 
duction of the £-trace is concerned, the step function has very little influence 
when {e/xoY ^ 1 (except that the interaction level F^ = is somewhat dis- 
placed). Writing z = x -^ jy we have 

^m {V) =a'y + /J-tan (y/xo) ^^ISmM 



cos (y/xo) + sinh {{x - €)/xo}' 
(39) 

/xt2n . ^ i If/ N / ^ cosh^ {(x — e)/Xo] 

(R. (r) = a-x + /3-tanh [i.x - .)/.„! - eoshMCx - ^ki - sin^ (,/.ro) ' 

If 5c is small we therefore have approximately 

22 = J-2/2 + Xa, = Xa, - J 



Ta„ I \ ^ 



jaoXo 1 + sinh- }(.r,, - t)/Xo] 



O. E. H. RYDBECK 



(S215) 151 



which means that 

(40) I Toz r = 



ToX^ = 0) 



1 + 



7a„Xo 1 + sinh {{Xa, - e)/xo} 



We see that only when (xa^ — e)/xo is small does the step function increase the 
transmission coefficient. Even so the resultant effect is significant only if Xq is 
small and at the same time (e/xo)^ <K 1. This means in practice that a step in 
the electron distribution function, in order to be effective in producing the 
2-wave, must occur almost at the regular interaction level Za^ and it must be 
quite steep at the same time. 





















1 


Direction of magnetic 


i^ein 










^ 


r 






I 


J 


^ 


x^ 












/ 


/ 


X 


/^ 




~o- 


< 




\ 

\ \ 






p^ 




1 




'/ 


yfh 


/ j 




\ 


\ 


Y 


\ \ 






1^ 








nh 


1 






\ \ 




J' ° 

e..77°3' 
H. 0,5/5 G 

\-aom 


\ qJ 


0.60 


o.e 


[,. 


^ 


m 


P 


/ 


/ 


m 


Li 




\ 


\\ 


\ 


J 


\\ 


IP / 


yj 


7 


/ 


II 


/ 






\ 


\ 




sV 






y 


/ 


/ 


/ 














/ 






"^1 




^ 










s,n 




■^^ 


^ 







INDEX OF REFRACTION FOR THE ORDINARY WAVE WHEN W-0 

VERTICAL I INODENCE 

Figure 16 



Let us next study if such a step, occurring practically at Za^ , is effective in 
noticeably increasing the transmission coefficient when Vc is very large, i.e. at 
low geomagnetic latitudes. Because v^ is very large, v <<^ v^ even for the lower 
ionosphere. We then have 



/bo / (^2 

Jzao 



rix) dz = j 






(712 — ^i) d/i 



+ 



7a„.To cosh (z/xo) 



assuming e very small. For a "strong" step function /3/.To » 7«o and ju ~ 
{^/8c) tan (z/xo). li dc^ ^ we find approximately 



152 (S216) ELECTROMAGNETIC WAVES 

I To, I' ^ exp {-2A:oXo[taii-' (5,//3) - tan"' (8/^)]} 
(41) 

^ exp [-koXoW - 28/^]] ^ exp {-koXoTr}, 5c » /S 

We have thus found that ''strong'^ steps in the electron density distribu- 
tion, located at the interaction level, can produce an interaction or coupling 
independent of large values of 5c . Unless the step is located at or very near 
the interaction level this effect mil not occur. 

Let[us study the situation briefly numerically. If we assume a geomagnetic 
latitude of about 30° and a wave frequency of t Mc/s, 8^ = 0.1. With 7o„ = 
0.1 /km (and neglecting /z) our basic formula for the smooth ionosphere jdelds 



To. \' ^ exp {-2ko8MO)/yaA ^ e"'" ^ e 



if NiO) = 0.5. 



In this case a ''discontinuity" or step would, therefore, theoretically produce a 
better 2;-trace (better | Toz |) only if Xq < 1/x km ~ 300 m = 3Xo . As the trans- 









OKI* 


Dirtetion of 








magnetic fleia 








^^^ 










^v 


/ 




^jjo 1^ 


\ \ P \ \ 


/ 


/ III II 

1 1 ~^~4-~~^i //' 


i / /^L 


^ 


S \ \ 


A- 




ml 




r \ \ 


4. .' 


/ .eo \~is~A.9B\h 


fill 


■SO SO 


^ 1 




I / ilili 












\ m 


ffJi.OB 1 


~/r^^^ / 


\ 






\m 


' 1 1 

11 1-30 1 


/ / ^/^^^--w 


\ 


\ I 


\ in 


If./ 


1 


\ 


\ \ \ \1L/ ^ 


1^/ 


/ / 


\ 




A '"^ 


/ / J 


©, = 77' ^ 




^y 1.0, y 


/ / 


H '0.515 


\. \. "■^ '^r^vrT/ 


^^ ^/^ 


/ / 


^'^'BOm 


^^--•-^'^s^ 


^ 


// / 




>v Con fact 


points FJ 


\ 


^^ 




/ 7r/3. e7^ 








INDEX OF REFRACTION FOR THE ORDINARY 


WAVE WHEN v>-.S(5*>° 








Figure 


17 







mission coefficient of the smooth ionosphere is almost infinitely small in the 
case discussed, one would have to require that a:o = 30 w in order to produce a 



O. E. H. RYDBECK 



(S217) 153 



significant result. A ''discontinuity" or step coupling therefore yields practical 
results only if the step takes place within about one vacuum wave length. This 
requirement, together with the condition that the step be located at the inter- 
action level would make the frequency of "step-excited" triple splits extremely 
small. It may, however, serve as an explanation of the extremely rare cases re- 
ported from more southern latitudes, provided the inospheric irregularities are 
sufficient. 

Finally, one is not surprised to find that exp { — koXoir } also is the amplitude 




/ 

I 



/NDEX or REFRACTION FOR THE THIRD (z-> COMPONENT WHEN \^ • .95 \f° 



Figure 18 



of the reflection coefficient of a medium where e = 1 + /3-tanh (z/xq), [11, p. 21], 
if i3 « 1, and the wave length is sufficiently short. 



6. Approximate Construction of the Ray Paths 

Many years ago Booker [13] gave a theoretical method of determining the 
ray paths of the ordinary and extraordinary rays. Here we find it convenient 



154 (S218) 



ELECTROMAGNETIC WAVES 



to use the graphical methods of crystal optics which have recently been used 
by Poeverlein [14] to determine ray paths in the case of zero losses {v = 0). 

As n is a function of V^ and cp, the angle that the wave normal makes with 
the vertical, the geometrical optics requires 



(42) 



n(V^, (p) -sin ^ = sin (pi 



We have thus to find the intersections between the n-curves, plotted as functions 
of coc/co^ and cp, and the line sin cpi as sho^vn in Figure 16 for Kiruna conditions 
for the ordinary ray when v = 0. The ray or energy direction is normal to the 
n-curves as shown by the small arrows. It is easily seen that the ray path is 




Fig. 19. Depicting the shape of the n(0, </>)- contact curves. 

no longer syrmnetric and the vertically incident ordinary ray is deviated towards 
the north. The magnitude of this deviation has been the object of a study by 
S. Forsgren of our laboratory [15]. 

In order to study the shape of the 2;-ray we have to plot the n(F", ^) -curves 



O. E. H. RYDBECK 



(S219) 155 



for V > 0. The ordinary and z (extraordinary) ''ellipses" now become very 
complicated on account of the fact that for v > v^ the o-wave transforms to a 
z-wave (longitudinal transmission). This is clearly shown by Figures 17 and 
18 for n{V^, (p)o and n{V^, (p)^ . At P the curves have contact points for V^ = 0. 
These are the points of practically complete or critical coupling between the o- 
and 2-waves. Here not only the wave normals but also the directions of energy 
flow are the same. 

The shape of the contact curves, n(0, (p), is depicted separately in Figure 18 
for the sake of convenience. If the inclination di is such that for vertical inci- 
dence Vc = V, our vertical line, for the construction of the vertical incidence ray 
paths, will pass through the two P-points above each other indicating critical 



}^m0.3 0^' 2.22- 10^ c/sec 




geomagnetic 
latitude es" 



Fig. 20. Ray paths of the o- and ^-components when v — 2.22 • lO^c/s for geomagnetic latitudes 
65° (critical coupling) and 50° (loose coupling). 

coupling and practically complete energy transfer for the up-going and for the 
do^vn-coming w^ave from z to o. Now even in the case of moderate losses, as 
in the present cases, the ray paths are quite accurately given by the previous 
geometrical methods [16], except near the strong coupling regions and the levels 
of total reflection where the geometrical methods never are applicable anyway. 



156 (S220) 



ELECTROMAGNETIC WAVES 



It is therefore quite easy, making use of Figures 17 and 18, to construct the o- 
and 2;-rays for vertical incidence. The result is shown in Figure 20 for vertical 
incidence and v = 2.22-10^ c/s at the geomagnetic latitudes 65° and 50°. The 
characteristic difference between critical and loose coupling is well demonstrated. 
One further finds, from Figure 19, that for angles of incidence less than 
v?oc , one obtains complete 0-2-0-rays. Examples of the limiting paths 
{(Pi = 0, (Pi = (poc north) are shown in Figure 21 for the geomagnetic latitude 
50° and, as before, v = 2.22-10^ c/s. The characteristic distortion of the paths 
is clearly shown. 



critical coupling 




magnetic field 



/ 


6.-67" 


/ 


H'OSIBG 


/ 


*» = 0.30 ^^ - 2,23 10^ c/sec 


/ 


v'^- 7.5 10^ c/sec 


/ 

/ 


^^[^).OiA.m 


^/oose /coupling 






km 

'S 

angle of incidence S'2 north vertical incidence 

geomogntfie letiiuac SO 

Fig. 21. Ray paths at the limiting oblique incidence, <poc for ;' = 2.22 • 10^ c/s and a geomagnetic 

latitude 50°. 



The limiting angle, (poc , normally is quite small. If di be the inclination 
that would yield v^ = v and di^ the actual inclination, one has 



(43) 
where 

and 



sm (poc = (Re fyip) 's'm (Si, — di), v > v^ 
(^e M = {^[^ + (1 + gYy2a + 9')V'\ 



= y-cos Ui . 



For di, = 67° (geomagnetic latitude 50°) and v = 2.22-10^ c/s, (poa is small or 
only 5.2°. 



O. E. H. RYDBECK 

7. Experimental Results 



(S221) 157 



One finds from Figure 14 that, at Kiruna for example, where the geo- 
magnetic latitude is about 65°, triple splitting cannot occur below about 120 km 
but that it must occur Avith considerable strength above that level. This is in 



PI 



Apr// 3, /9^9 19^3 GMT 



Virtual 
he/g/it 
/<m 




Frequency t^cjs 



Fig. 22. jEJ-layer penetration triple splitting recorded at the Kiruna 
Observatory, April 3, 1949, 1543 GMT. 



August /Z /9S0 /^36 GMT 



V/rtua/ 
tie/g/it 
/<m 




Frequency Mcjs 

Fig. 23. E'-layer penetration triple splitting recorded at the Kiruna 
Observatory, August 17, 1950. (Note the F triple split also recorded.) 

complete agreement with our experience at this northern observatory where 
triple penetration of the £'-layer is recorded almost regularly. In Figure 22 is 
shown one of the typical records obtained with the 16 kW panoramic recorder 



158 (S222) 



ELECTROMAGNETIC WAVES 



(running without the anti- jamming circuit at the time in question). It is in- 
teresting to note that, on this occasion, as on so many others, the o-z-o-com- 
ponent is about as strong as the 0-component. 



Virtual 
height 




I 
I 

I 



/ 



7^ Angular wave 
frequency 

Fig. 24. z-, o- and x-traces for smooth E- and F-Iavers. 



October 6, 19^9 0AA2 GMT 



Frequency Mc/i 



Virtual 
height 
km 




Fig. 25. F-layer triple splitting recorded at Kiruna October 6, 
1949, 0442 GMT. 

If the recorder were moved to an even higher geomagnetic latitude the 
o-2-o-component would be the strongest one. Finally, one notes from Figure 22 



O. E. H. RYDBECK (S223) 159 

the characteristic and obvious fact that the o-^-o-wave, reflected from the F- 
layer, disappears when the o-wave penetrates the £^-layer. This feature of the 
2-trace has already been reported by Meek [4]. On certain occasions, however, 
when the conditions are favourable, the sensitive recording equipment will 
actually show that this 2;-trace is delayed when the o-wave penetrates. Figure 
23 is a recent example of this delay obtained at the Kiruna Observatory. 

Figure 24, a graphical sketch of the virtual heights for a smooth E- and 
F-layer, serves as an additional illustration of the situation. 

In Figure 25 we have reproduced a typical average record of a triply split 
F-echo from Kiruna. The F-triple splits are somewhat less frequent than the 
^/-triple splits at Kiruna as already mentioned. 

8. Concluding Remarks 

The present investigation has shown that with respect to the 0- and z- 
waves the layers of the ionosphere act as if they constituted coupled trans- 
mission lines. The ''ordinary" lines run from Zq — Ah to Za^ and from 2zo — 0„„ 
to Zo + Ah for a symmetrical layer. The "z^^ line, adjoining the two ''ordinary" 
lines, runs from Za^ to 2zo — Za^ (Zao < ^o). When Za^ = Zq , i.e. for wave fre- 
quencies above the ^-critical frequency, the layer acts as one "ordinary" line 
and the 2-trace disappears. 

Obviously it is not possible to detect any difference in polarization between 
the 0- and o-;s-o-waves outside the layers. There they are, in fact, both ordinary 
waves. 

The author's thanks are due to the Swedish Research Councils for the Technical and 
Natural Sciences for grants which made this investigation possible. The excellent assistance is 
acknowledged of S. Forsgren, E. E.,who prepared most of the graphs of this communication. 



Bibliography 

1. Eckersley, T. L., Discussion of the ionosphere, Proceedings of the Royal Society, Series A, 

Volume 141, 1933. 

2. Toshniwal, G. R., Three-fold magneto ionic splitting of the radio echoes reflected from the 

ionosphere, Nature, Volume 135, 1935, 

3. Harang, L., Vertical movements of air in the upper atmosphere. Terrestrial Magnetism and 

Atmospheric Electricity, Volume 41, 1936. 

4. Meek, J. H., Triple splitting of ionospheric rays, Nature, Volume 161, 1948. 

5. Newstead, G„ Triple magneto ionic splitting of rays reflected from the Fi region, Nature, 

Volume 161, 1948. 

6. Seaton, S. L., Magneto ionic refraction at high latitudes, Institute of Radio Engineers 

Proceedings, Volume 36, 1948. 

7. Rydbeck, O. E. H., The ionospheric and radio wave propagation observatory at Kiruna, 67° 50' 

N, S0° 14. 5'E, Tellus, Volume I, No. 4, 1949. 

8. Rydbeck, O. E. H., Magneto ionic triple splitting of ionospheric waves, Proceedings of the 

Indian Academy of Sciences, Volume 31, No. 2, 1950. 



160 (S224) ELECTROMAGNETIC WAVES 

9. Rydbeck, O. E. H., On the propagation of radio waves, Transactions of the Chalmers Uni- 
versity of Technology, Gothenburg, Volume 34, 1944. 

10. Appleton, E. V., and Builder, G., The ionosphere as a doubly refracting medium, Proceedings 

of the Physical Society^ Volume 45, 1933. 

11. Rydbeck, O. E. H., On the propagation of waves in an inhomogeneous medium. Report No. 7, 

Research Laboratory of Electronics, Transactions of the Chalmers University of 
Technology, Volume 74, 1948. 

12. Eckersley, T. L., Coupling of the ordinary and extraordinary rays in the ionosphere, Proceed- 

ings of the Physical Society, Volume 63, No. 361B, 1950. 

13. Booker, H. G., Oblique propagation of electromagnetic waves in a slowly-varying non-isoiropic 

medium. Proceedings of the Royal Society, Series A, Volume 155, 1936. 

14. I^oeveAem,}!., Strahlwege von Radiowellen in der lonosphdre, Sitzungsberichte der Bayeri- 

schen Akademie der Wissenschaften, 1948. 

15. Forsgren, S., Some calculations of ray paths in the ionosphere. Report from the Research 

Laboratory of Electronics 1950, to appear in the Transactions of the Chalmers 
University of Technology. 

16. Rydbeck, 0. E. H., A theoretical survey of the possibilities of determining the distribution of 

the free electrons in the upper atmosphere, Transactions of the Chalmers University 
of Technology, Volume 3, 1942. 



An Asymptotic Solution of Maxwell's Equations* 

By MORRIS KLINE 
New York University 

1. Introduction 

In view of the difficulty of obtaining exact solutions of Maxwell's equations 
under given initial and boundary conditions and the difficulty of obtaining 
practically useful solutions even where exact, explicit solutions are known the 
potentialities of asjrmptotic solutions warrant investigation. This paper derives 
a form of asymptotic expansion suited to initial and boundary conditions shortly 
to be specified and then shows how it is at least theoretically possible to determine 
the successive coefficients of the expansion through the solution of ordinary 
differential equations/ 

A somewhat more specific discussion of the material of this paper follows. 
Through a form of DuhameVs principle we can relate the electromagnetic field 
due to an arbitrary electric charge distribution with harmonic time behavior to 
the field created by the same charge suddenly placed in space at time t = 0. 
The latter field, denoted by Eq , Ho , is to be called the pulse solution of Max- 
well's equations. Both fields are required to satisfy the initial condition of 
being zero for t < and both are required to satisfy Maxwell's equations for 
the same electromagnetic parameters e, ijl, and a, the latter being assumed to 
be sectionally continuous functions of x, y, and z, thereby allowing for abrupt 

Paper presented at the June, 1950, Symposium on the Theory of Electromagnetic Waves, under 
the sponsorship of the Washington Square College of Arts and Science and the Institute for 
Mathematics and Mechanics of New York University and the Geophysical Research Direc- 
torate of the Air Force Cambridge Research Laboratories. 

^The form of the asymptotic expansion given in this paper and its derivation from 
Duhamel's principle were presented by R. K. Luneberg in a series of lectures given at New 
York University during the academic year 1947-1948. Luneberg derived Duhamel's principle 
for Maxwell's equations by Laplace transformation of the equations. In this paper the prin- 
ciple is verified directly. Those portions of this paper which repeat Dr. Luneberg's work are 
presented here for completeness because the material was not published by Dr. Luneberg before 
his death in 1949. Work on the new material of this paper, which concerns the derivation of the 
ordinary differential equations for the coefficients of the expansion from discontinuity condi- 
tions, was begun by Dr. Luneberg but left incomplete by him. 

*The work on this paper was done at the Washington Square College of Arts and Sciences 
of New York University and was partially supported by Contract No. AF-19(122)-42 with the 
U. S. Air Force through sponsorship of the Geophysical Research Directorate. Air Force 
Cambridge Research Laboratories, Air Materiel Command. 

225 (S225) 



226 (S226) ELECTROMAGNETIC WAVES 

changes in media. It is further assumed that any discontinuities with respect 
to the time variable in Eq , Ho , and their successive time derivatives are finite, 
an assumption fulfilled in numerous physical situations. 

From Duhamel's principle it is possible to show first that the field created 
by the time harmonic source approaches with increasing time a field ha^dng 
the harmonic time behavior; that is, except for what may be called a transient, 
the field due to the harmonic source approaches the form u exp { —ioit], v exp 
{ —icx)t}, where u and v are vector functions of x, y, z, and co is the circular fre- 
quency of the harmonic source. 

The asymptotic expansions which are the subject of this paper furnish 
expressions for u and v in the form of series in both of which the basic variable 
is the reciprocal of co, or, alternatively^ the wave length X. The n-th coefficient 
of the series for u is essentially the sum of the discontinuities with respect to 
the time variable of the {n — l)-st time derivative of Eq . A similar statement 
applies to v and Ho . These expansions for u and v are noteworthy partly 
because for the limiting case of X approaching zero they give the geometrical 
optics approximation to the time harmonic field. 

Since the asymptotic expressions for u and v can be obtained from certain 
discontinuities in the pulse solution one might logically seek that solution. In 
general, however, the problem of obtaining the pulse solution corresponding to 
the initial conditions and the conditions on e, fi, and a is as difficult as sohing 
for the field due to the time harmonic source. But the asymptotic expansions 
depend only upon certain discontinuities of the pulse solution and this fact 
suggests the problem of determining these discontinuities directl}' without re- 
quiring a knowledge of the full pulse solution. 

To determine the discontinuities directly we first recast ^Maxwell's differ- 
ential equations into the form of integral equations which admit a class of 
discontinuous solutions. Conditions on the discontinuities of these solutions 
are then obtained which take the form of a recursive system comprising both 
linear algebraic and linear partial differential equations. This system is recast 
into a system of recursive ordinary differential equations for the discontinuities 
of the successive time derivatives of any admissible solution, E, H of the in- 
tegral equations. Further conditions in the form of initial conditions on the 
solutions of the ordinary differential equations are introduced to insure that 
these solutions give the discontinuities of the pulse field rather than those 
belonging to some other field. 

The entire theory offers a method of obtaining asjonptotic solutions for 
some classes of electromagnetic problems. 

2. Mathematical Statement of the Problem. 

We consider, to begin with, the problem of solving IMaxwell's equations in 
a general non-homogeneous isotropic medium, which, for simplicity, is assumed 
to have zero conductivity. We shall write Maxwell's equations in the form 



MORRIS KLINE (S227) 227 



(la) curl K--E, =-¥, 

c c 



(lb) curlE + ^H, = 0, 

c 

wherein e and /x are sectionally continuous functions of x, y and z. The real 
part of (l/47r)F< represents the enforced current density due to some distribution 
of charge whose strength varies with time and is to be distinguished from the 
usual current term ctE which represents induced current due to free electrons in 
conducting media. The function F(.t, y, z, t) shall be considered as given and 
determines, as we shall see, the charge distribution which creates the field E 
and H. The components of F are assumed to be sectionally smooth functions 
of X, y, z, t. It is understood that F = for ^ < so that the source begins to 
act at i = 0. 

The physical meaning of F and the justification for including it in equations 
(1) may be seen by taking the divergence of both sides of the first equation. 
Since div/curl = 0, we have div eE, = — div F« . If we assume for the moment^ 
that integration with respect to t introduces no new function of x, y, z then 
div eE = — div F. However, by one of the well known electromagnetic equa- 
tions, div D = 47rp where p is charge density. Hence 47rp = —div F and it is 
clear that F determines the charge density. From this last equation and the 
divergence theorem it follows that jv 47rp dv = —j^ ¥-n da. This relation 
justifies describing F as the flux of source charge. 

From 47rpt = —div Ft and the equation of continuity, V-J = — p* , we 
have V- J = (l/47r)V-Fi , which means that (l/47r)F« = J = current density, 
except possibly for some divergenceless function of x, y, z. Such a function would 
have to represent a divergenceless current density which on physical grounds 
can be assumed not to exist at x, y, z where the source current density F, exists.^ 

We shall generally write F as g(x, y, z)f(t) where g is a vector function 
and /, a scalar, both possibly complex. Such a form for F is physically reason- 
able since it means merely that the charge remains at a fixed location but varies 
in strength with time. Also if /(O should be the Heaviside unit function 7](t), 
that is, for i < and 1 iov t > 0, then the current source in Maxwell's equa- 
tions is a pulse of current which has infinite strength at ^ = but is non- 
existent for other values of t. The field which arises from such a current pulse 
will be called the pulse solution of Maxwell's equations and is denoted by Eq , 
^0 . Such a field will of course spread out into space from the region in which 
the source charge exists and mil approach with increasing time the static field 



^See the proof in section (4) after equations (21) and (22). 

^If the term crE had appeared on the right side of the first of Maxwell's equations, all of the 
above arguments as to the meaning of F would apply provided merely that F exists at points 
in space at which o- = 0. This is a reasonable assumption since sources are generally placed in 
non-conducting media. 



228 (S228) ELECTROMAGNETIC WAVES 

created by a fixed charge having the same strength and geometrical distribution. 
In particular Ho(a:, y, z, «>) will be zero. Since F = for ^ < it is also true 
that Eofc y, z, 0-) = Ho(x, y, z, 0-) = 0. 

Fo' a time harmonic source we shall let J(0 be exp { —iwt] for ^ > and 
for t < D. Here too the resulting field will be zero for t < 0. 

The function g which determines the spatial distribution of charge may 
be accommodated to various physical sources. For a Hertzian dipole g = 
M6(x. 2/, z) where M is the constant vector moment of the dipole and b{x, y, z) 
is the Dirac 5-f unction which is infinite at x, y^ z and zero elsewhere. 

One can now determine a solution of Harwell's equations for a given F 
under the given initial conditions that E(x, y, z, t) = H(x, y, z, t) = for t < 0. 
However, this solution need not be unique partly because discontinuities in F 
are permitted. Discontinuities in F mean discontinuities in E and H at ^ = 0+, 
for E and H are zero at ^ = 0+ where the g{x, y, z) in. ¥ = gf is zero, and E 
and H are not zero at t = 0+ where g ?^ 0. Under discontinuous initial condi- 
tions on E and H it is possible to show by examples that the resulting solutions 
of Maxwell's equations need not be unique. In addition, discontinuities in e 
and fjL, which mean a change in medium, call for the imposition of conditions 
which relate solutions of Maxwell's equations in the differing media. We shall 
therefore shortly obtain conditions on the discontinuities in E and H which are 
to be satisfied wherever and whenever such discontinuities occur. Under these 
added conditions the solution is presumably unique.* 

The problem to which we address ourselves is to find solutions in asjTQptotic 
form for Maxwell's equations. These solutions will be fields created by time 
harmonic sources, are to satisfy the initial condition that E and H are zero for 
^ < 0, and must correspond to the given, sectionally continuous e and ju. Since 
w^e shall shoAV shortly that apart from a transient quantity, that is, a quantit}^ 
which approaches at each x, y, z with increasing time, the fields are also 
time harmonic, and indeed possess the same circular frequency as the source, 
E and H will have the form u exp { —icot] and v exp { —ioit} where u and v are 
complex vector functions of x, y, z. We shall therefore seek asjTQptotic expres- 
sions for u and v, which give the spatial behavior of the time harmonic part of 
E and H. When these asymptotic expressions are obtained we shall offer a 
method of determining the coefficients of these asymptotic expressions. 

3. Derivation of the Asymptotic Expressions from DuhameVs Principle 

Let Eo , Ho be the pulse solution of Maxw^ell's equations for given e and n 
and spatial charge distribution g. Let E, H be the solution corresponding to 
the same e, ijl, and g but for arbitrary time behavior f{t) of the charge. Both 



^See the discussion of equations (12) in section 4. 



MORRIS KLINE (S229) 229 

solutions are to satisfy the same initial conditions, that is, for i < 0. Then 
the form of DuhamePs principle which we shall employ states that 

(2a) E = 1^ £ Eo(^ - T)f{r) dr 

(2b) ^ = itIo ^'^^ ~ ^^-^^^^ ^^' 

While a deductive proof starting from Maxwell's equations and leading to 
Duhamel's principle can be given it is sufficient for the purposes of this paper 
to verify the correctness of the principle by direct substitution of E and H in 
Maxwell's equations. 

One point about this verification needs special attention. On physical 
grounds it is clear that Eq and Ho are certainly discontinuous in the time variable, 
for, suppose that at ^ = a charge located at some position is suddenly allowed to 
act. At this time and at any point (x, y, z) distant from the charge the field is 
zero. However, at some later time, ty say, the field created by the charge will 
reach (x, y, z) and a sudden change will take place there in the value of Eo and 
Ho at time ^i . Thus Eq , Ho , and their successive time derivatives will be dis- 
continuous at ^ = ^1 . If e and /x are discontinuous along some surface, then the 
field upon reaching that surface will be partially reflected and the reflected 
field may pass (x, y, z) at some later time, ^2 say, at which time another dis- 
continuity will exist in the values of Eo , and Ho , and their successive time 
derivatives. In addition, discontinuities in Eo and Ho will exist at time t = 
at those {x, y, z) at which charges are placed, for at i = — , the field is but 
at t = 0+ the field created b}^ the charge arises immediately. 

In verifying Duhamel's principle it is therefore necessary to take into 
account discontinuities in Eo and Ho . Let (xi , yi , z^ , ti) be a point at which 
Eo and Ho are discontinuous and let t^ be such that < ^1 < ^. The verification 
is accomplished by showing that the indefinite time integrals of E and H satisfy 
the time integrals of Maxwell's equation. We consider the first of IMaxwell's 
equations and write 

curl f Ildt--E = -¥. 
J c c 

We may now substitute (2a) and (2b) in this equation to obtain 

(3) curl f Ho(< - T)fiT) dT--^l' Eo(^ - r)f{r) dT = -¥. 

Jo C Ot Jo C 

Consider the second integral. Since Eo{t) is discontinuous at / = ^i , Eo(^ — r) 
is discontinuous for r = ^ — ^1 . Let us break up the r-interval of integration, 
to t, into to ^ — ^1 and t — U to t. Within these subintervals Eo(^ — t) is a 



230 (S230) ELECTROMAGNETIC WAVES 

continuous function of r. Moreover we may suppose on physical grounds that 
Eo and dEo/dt are continuous dXt = t. Hence^ we may differentiate the separate 
integrals ^dth respect to t and obtain 

-; /" E„,« - T)/(r) dr - ; E„«t).f« - U) - \ Eo(0)/(() + ; E„(Q/(( - « . 

We consider next the first integral in (3). Since the curl operation also 
involves a set of differentiations under the integral sign we must again consider 
the discontinuity in Ho(0, which we may with no loss in generahty suppose to 
occur 2Xi = U . We must now note that in later applications U will be a function 
of X, ?/, z and we may write the equation of the hyper-surface on which the dis- 
continuities X, ?/, 2, i lie in the form tx = ^(x, y, z)/c. We again break up the 
T-range of the integral into to t — ti and t — U to t. Considering now only 
one of the differentiations involved in the curl operation and designating by 
Ho3 the third component of Ho , we have 

i f Ho3(^ - .)/(r) dr 



£ ^ Ho3(^ - T)}{r) dr - Hos{t\)f{t - « f^ + Ho3(/T)/(/ - ^0 



dlf 



If we now take into account all of the terms involved in the curl operation we 
obtain 

curl [' Ho(^ - r)fir) dr 

= f curl Ho(^ - r)f(r) dr -- grad rp X lB.o{Q]f{t - Q, 

Jo ^ 

wherein [Ho(^i)] equals Ho(^i+) — Ho(^i — ). 

If we substitute the two results of differentiation in (3) and use the fact 
that Eq and Ho are a solution of Maxwell's equations for the pulse source F = 
gry we obtain 



/. 



l ei'it - t)/(t) dr 
C 

- grad ^p X [Ho(«]/(/ - « - - [Eo{h)]f{t - U) - l Eo(0)/(0 = j^ gf{t). 

C COL- 



=See P. Franklin, Treatise on Advanced Calculus, Wiley, New York, 1940, p. 348. We are 
supposing, too, that the discontinuity in Eo/(0 at t = tiis finite. This assumption is consistent 
with later work in this paper. 



MORRIS KLINE (S231) 231 

We shall show later' that Eo(0+) = -g/e. Since rj'it) = ior t > 0+, the 
integral is 0. We therefore arrive at the condition 

grad ^p X [Ho] + 6[Eo] = 

on the discontinuity hypersurface ^i = xp/c as the condition that the E and H 
given by (2a) and (2b) satisfy the first of Maxwell's equations. 

A similar discussion to show that this E and H satisfy the second of Max- 
well's equation would lead to the condition 

grad ,A X [Eo] - /x[Ho] = 

on the hypersurface ^i = yp/e. 

We require that Eo and Ho satisfy the above conditions on a discontinuity 
surface and the verification of Duhamel's principle is thereby completed. These 
conditions are in no sense a limitation on Duhamel's principle since some condi- 
tions must be placed on Eo and Ho to relate solutions of Maxwell's equations 
across a discontinuity surface. We shall see later that the above conditions are 
precisely the ones we must impose. 

We may now derive from Duhamel's principle the asjnnptotic expansion 
with which this paper deals. Since the expansion applies only to fields created 
by time harmonic sources we shall now let /(Q = exp { —iwt]. Let t — t = s 
and t = tin (2a) . By this change of variable we obtain 

E = — / Eo(s) exp {—ico{t — s)\ ds. 

ot Jo 

Taking the exp { —imt] factor outside the integral and differentiating the product 
gives 

E = Eo(0 — ^a) exp {—ioit} j Eo(s) exp {^cos} ds. 

Adding and subtracting equivalent terms gives 
E = Eo(0 - Eo(-) 

— exp {— zco/n — Eo(oo) + ^co / (Eo(s) — Eo(oo)) exp {icos] dsf. 

This result shows that with increasing time E{x, y, z, t) approaches^ the 
form u exp { —ioit} where, changing s to r in the preceding result, 

(4a) u(x, y, z) = Eo(oo) - ico / (Eo(t) - Eo(«>)) exp {fcor} dr. 

Jo 

^See footnote 11. We also show later that Ho (0+) =0. This fact is needed in verifying 
that E and H satisfy the second Maxwell equation, 

^The existence of the infinite integral in (4a) may be assumed on the ground that Eo(t) 
approaches, with larger and larger r, the field of a static charge and hence the integrand must 
approach zero sufficiently rapidly for convergence. 



232 (S232) ELECTROMAGNETIC WAVES 

That is, a time harmonic source gives rise, except for a transient, to a time 
harmonic field of the same frequency. 

Similarly, H approaches the form v exp { — icct ] where 

(4b) y(x, y, z) = Ho(cx.) - ice [ (Ho(t) - Ho(oo)) exp liccr] dr. 

Now integrate (4a) by parts, letting the u of the parts formula be Eo(t) — 
Eo(oo) and the dv be too exp {iur] dr. Then 

U = Eo(oo) - (Eo(t) - Eo(oo)) exp {icer} + Eo.(r) exp {io^rl dr. 

lo «-'o 

In evaluating the second term on the right side we must take into account the 
fact that Eo(r) is discontinuous. Let t^ , t2 , • • • , r^ , • • • , r„ be values of r 
at which Eo(r) is discontinuous.^ Further, let [Eo]a = Eo(t„ +) — Eo(ra — ). 
Then 

U = Eo(0+) + Z [Eo]a exp (icorj + [ Eo.(t) exp [iccr] dr. 

We again integrate by parts obtaining 
u = Eo(0+) + S [Eo]exp {zcorj 

a 

- — X) [Eorla exp {icoT^} - — / Eo.. exp {tor} c?t. 

The second summation includes strictly a discontinuity at r = and 7=00. 
However, we shall show^ later that except at special points Eor(0+) = 0; also 
we may assume that Eor(°°) = on physical grounds since Eo(t) approaches 
the field of a static charge as r becomes infinite. The same remarks apph' to 
the higher derivatives of Eo(r). Eo(0+) has already been noted to equal — g/e 
and so may be replaced by that quantity. 

It is apparent that the process of integration by parts can be continued as 
long as the next higher time derivative of Eq exists and as long as the discon- 
tinuities of the time derivative of Eq which appears in the integrand are finite. 
These conditions are certainly fulfilled for large classes of physical problems. 



8Eo(t) may be discontinuous at an infinite number of values of r. In such cases the sum- 
mations which occur in the succeeding steps must be taken over the infinite number of values 
of a. On physical grounds these sums may be expected to converge, for the physical circum- 
stance under which an infinite series of discontinuities occurs is that of a series of reflections of a 
family of wave fronts running between two refracting surfaces, e.g., internal reflections in a 
lens. At each reflection the value of Eo and hence the next discontinuity in Eo which occurs at 
(x, y, z) may be expected to decrease sufliciently rapidly to insure convergence. 

^See footnote 11. 



MORRIS KLINE (S233) 233 

We may therefore write 

U = -g/e + 2 [Eo]a exp {zcorj 



(5) 






The summation which appears in each term must cover all the discontinuities 
which exist for the derivative in question. 

The infinite series in (5) is truly an asymptotic expression for u, for, the 
remainder after n terms is readily seen, by an integration by parts, to be of 
order one higher in 1/io) than the n-th term. On the assumption that the integral 
which results from the integration by parts remains finite as co becomes infinite, 
the definition of an asjrmptotic expansion is satisfied. In view, however, of the 
physical meaning of Eo(r) the time derivatives may be expected to approach 
zero sufficiently rapidly. 

The series (5) can be given a slightly altered form. The points {x, y, z, t) 
at which Eo , Ho , and their successive time derivatives are discontinuous in t 
may be regarded as lying on some complicated hypersurface </)(x, y, z, t) = 0. 
We assrnne on physical grounds that it is possible to solve for t. This solution 
may have many branches, t^ = ^a(x, y, z)/c. As a matter of fact, at points 
where e, ju, and F are continuous we shall show later that the xpa are wave-fronts, 
that is, they satisfy the eiconal equation i^x + ^y + 'A! = €/i. Now, in (5), x, y^ 
and z are suppressed variables. For any given {x, y, z) at which u is being 
evaluated, there are various values ta at which Eq or some time derivative is 
discontinuous. At that x, y, z the ta are given by xpaix, y, z)/c. Hence we 
may write (5) as 



(6a) 



u = -g/e + E [Eo]. exp l^k^p^} - I J2 [Eo.]« exp {ikrP. 



+ 7^2 X [Eorrla eXp {Ik^ a} - '" 



By the same argument we obtain the asymptotic expansion for v except 
for the fact'*" that Ho(0+) = 0. Hence 

V = 2 [Hole exp [ik\p^} - ~ Yl [Ho.]a exp {ikxp^} 
(6b) 

+ T~^ Z) [Horrla CXp {Ikxp ^] - '" . 



loSee footnote 11. 



234 (S234) ELECTROMAGNETIC WAVES 

One point about the expansions for u and v that is especially noteworthy 
is the significance of the leading terms. Ignoring the term — g/e in (6a) since 
this term is zero except at points where source charge exists, we note that for 
CO — > oo or X — ^ the first term in each expansion should be a good approximation 
to the spatial behavior of the time harmonic field. These first terms are the 
approximation given by geometrical optics. 

Formulas (6a) and (6b) constitute a method of finding the asymptotic ex- 
pansion of a large class of time harmonic fields. In particular they apply to 
those physical situations where discontinuities in Eq and Ho arise from the sudden 
passage past any point x, y, z of wave fronts created by the source and coming 
either directly from the source or reflected from other discontinuities in the 
medium. In these situations the various assumptions made above, and there- 
fore the asymptotic expansion, apply. 

One can use formulas (6) if the pulse solution Eq , Ho is kno^vn. It is 
possible to obtain Eo and Ho in some classes of problems, but, generally, finding 
Eg and Hq is as difficult a problem as finding E and H, or u and v, directly. 
However, formulas (6) show that the coefficients of the asjTQptotic expansion 
depend directly upon the behavior of Eo and Hq only in the immediate neighbor- 
hood of the discontinuity surfaces ta = ^a{x, y, z)/c. Hence the problem sug- 
gests itself to determine these coefficients directly instead of finding the complete 
Eo and Ho . The coefficients we seek are essentially discontinuities of a solution 
of Maxwell's equation and we shall therefore investigate these discontinuities. 
The procedure we shall adopt is to investigate the discontinuities of a large 
class of solutions which includes pulse solutions and we shaU obtain conditions 
which discontinuities of E, H, and the time derivatives of E and H must satisfy. 
These discontinuity conditions will then be used to determine the coefficients 
of the expansions for u and v. 

4. Derivation of the Discontinuity Conditions 

We shall be interested in fields E and H which are discontinuous along some 
hypersurface = 0, w^hich satisfy Maxwell's equations on either side of this 
surface, and which satisfy given initial conditions. Discontinuous solutions 
arise because the initial conditions are discontinuous and discontinuities in the 
medium are permitted. Apparently some conditions must relate the two 
fields which obtain on either side of the discontinuity surface else no unique 
solution is determined by the initial conditions. 

To treat discontinuous solutions of Maxwell's equations we shall replace 
these equations by integral equations in which E, H, and F, but not their de- 
rivatives, will appear. We shall then treat solutions of these integral equations 
and, in particular, derive discontinuity conditions from them. Further ad- 
vantages which result from working with the integral equations will be discussed 
later. 



MORRIS KLINE (S235) 235 

Consider first any region G of x, y, z, t space in which E, H, and all their 
derivatives exist. Let Q{x, y, z, t) be any scalar function which is continuous 
and has continuous partial derivatives of any finite order in G and which is 
identically on the boundary and outside of G. Let us multiply each of the 
equations (1) by 12 and integrate over G. If for brevity we let 

(7) C=^^E + iF, 

we obtain 

(8a) f {Q curl H - 12C,) dio = 0, 



(8b) 



J f 12 curl E + 12 - H J dw = 0, 



where by fo we mean the four-fold integral and dw = dx dy dz dt. 

Now, using Hi , H2 , H^ for the components of H and literal subscripts to 
indicate partial derivatives, we have for the first term in (8a), 

(9) / 12 curl H dw = j [ 12(^^3 ^ — H2^)^ + corresponding terms] dw. 

We apply integration by parts to each term with respect to the variable appear- 
ing in the partial derivative. Let F be the boundary of G. Then the integration 
by parts applied to the first term gives 

/ 12/^31, dw = / I2F3 dx dzdt - \ 12,^3 dw. 
Jo Jt Jg 

The Jt denotes that we must replace the independent variable ?/ in 12 and H3 
by h{x, z, t) and then by k(x, z,t),h andk being the equations of the two parts 
of r which bound G, and then subtract the result of the second substitution 
from the first. However, since 12 is on the boundary of G the triple integral 
vanishes. Application of this procedure of integration by parts to each term 
of (9) yields 

(10) / 12 curl Udw = - grad ^ X U dw, 

J G J G 

where the curl gradient operators are the usual three-dimensional operators. 

Next consider the second term of (8a). The same procedure of integration 
by parts, this time with respect to the variable t, yields 

(11) I ^Ct dw = - I 12,C dw. 

J G J a 



236 (S236) ELECTROMAGNETIC WAVES 

From equation (la) we have therefore derived the integral equation 

(12a) / (grad 12 X H - Q,C) dw = 0. 

Jo 

Similarly from equation (lb) we obtain 

(12b) [ (grad 12 X E + 12,(m/c)H) dw = 0. 

Jg 

Now in a region G in which E, H, their first order derivatives, F, and Ft 
exist the equations (1) are equivalent to equations (12), for not only have we 
derived (12) from (1) but we may derive (1) from (12) by proceeding first to 
(8) and then arguing that since (8) must hold for any function Q of the kind 
described above, the factor multiplying 12 in the integrand must be 0. 

If, however, we consider a region G in which the first order derivatives of 
E and H are discontinuous and possibly E, H, F, and F^ likewise discontinuous, 
then equations (1) do not hold throughout the region whereas equations (12) 
do still have significance. Hence we agree to replace ]Maxweirs equation (1) by 
the integral equations (12) for all future work in this paper, the purpose of this 
move being to admit discontinuous solutions of Maxwell's equations. 

One further condition will be attached to the use of the integral equations 
(12). Suppose that in the region G, the functions E, H, F or their derivatives 
are discontinuous at points {x, y, z, t) which lie on some hj'persurface = 
contained or partially contained in G (see Fig. 1). We shall suppose that E, H 
and F satisfy Maxwell's equations, and in fact have higher derivatives '^'ith 




Figure 1 



respect to t, in and on the boundary of Gx and Ga separately, the values of E, 
H, F and their derivatives on the boundary being the limiting values approached 
by these functions from the interior. 



MORRIS KLINE (S237) 237 

Imposition of the requirement that solutions of the integral equations (12) 
valid over the domain G also satisfy Maxwell's differential equations in Gi and 
(j2 separately raises the question of whether such solutions exist at all. It can 
be shown mathematically that if a discontinuous solution E and H is the limit 
of a sequence of continuous solutions E„ , H^ then this discontinuous solution 
will satisfy the integral equations (12) in G and Maxwell's equations (1) in Gi 
and G2 separately. That is, there are solutions which meet our conditions and 
it is for such solutions that we shall obtain discontinuity conditions. 

There is good physical reason to expect that discontinuous solutions are 
limits of continuous solutions. For example, a sudden change in the physical 
characteristics of a medium, which is idealized mathematically as a discon- 
tinuous change in the characteristics, can be replaced by a rapidly varying 
continuous change to which a continuous solution would correspond. As the 
continuous change is made more and more rapid so as to approach the sudden 
change, the corresponding continuous solutions of the field equations should 
approach the discontinuous solution corresponding to the sudden change. Hence 
one expects discontinuous solutions of the field equations to be limits of con- 
tinuous solutions. 

There are several advantages to replacing equations (1) by the integral 
equations (12). First, we have generalized Maxwell's equations so that we may 
take care of discontinuities in E, H, F, and their derivatives. Equations (12) 
hold as long as the components of E, H, and F are integrdble. Secondly, we have 
reason (though no rigorous proof) to expect the uniqueness of the solutions to 
equations (12) under the initial conditions stated in article 2, for the solutions 
we consider are limits of continuous solutions and the latter are unique for 
continuous initial conditions. 

A third advantage of equations (12), which at the moment is incidental, 
is that it permits us to treat without difficulty point source functions of the 
form F = M 5(x, 2/, ^)f{t), Avhere M is a constant vector and 8 is the Dirac delta 
function. For this F, using a well known property of the delta function, equa- 
tion (12a) reads 



[ [grad 12 X H - - 12,E] dw = ~ f 

Jg C C Jo 



12,(0, 0, 0, t)f{t) dt. 



We shall now apply equations (12) to a domain G within which discon- 
tinuities occur. Let 0(x, y, z, t) = be any surface dividing the region G (see 
Figure 1) into regions Gi and G2 and such that F, E, H, or some of their deriva- 
tives are discontinuous on this surface. However, in Gi and G2 and on the 
boundaries as approached from Gi or G2 let E, H and F have derivatives of 
any order. 

We may regard the fields E and H which are defined in G to consist of 
two fields. In Gi , E and H are to be the fields E^^^ and H^^^ which have the 
same values as E and H in Gi and which are in G2 . On = 0, E^^^ and H^^^ 
have the values approached by E and H from Gi . Like\vise in G2 , E and H 



238 (S238) ELECTROMAGNETIC WAVES 

are to be replaced by E^^^ and H^^^ which are to have the same values as E 
and H there but which are zero in Gi . Likewise, on = 0, E^^^ and H^^' are 
to have the values approached by E and H from G2 . With this understanding 
about the values of E and H in Gi and G2 we may now state instead of (12a), 
for example, 



(13) 



/ (grad 12 X H - 12,C) dw 

= I (grad 12 X H - 12,C) dw + f (grad 12 X H - 12,C) dw. 



The same applies to equation (12b). 

Let us now assimie that we have a field E, H which is a solution of equations 
(12) but which is discontinuous on the hypersurface = separating G into 
(ji and G2 . We shall apply equations (12) in the form indicated by the right 
side of equation (13) to derive conditions on the discontinuities on E and H. 

We first introduce the unit vector N which is normal to the surface = 0. 
The components of N are 

(14) N = ±X(0. , 0. , 0. , 00 

where X = 1/(0^ +0^ + 0^+ 0?)^^^? the plus indicating that N points into G2 
and the minus that N points into Gx . Now consider, as in the derivation of 
(10), a typical term of /©, grad 12 x H dw. Such a term is fo^ 9-yHs dw. By 
integration by parts with respect to y we obtain that 

/ 12^^3 dw = / 127^3 dx dzdt - \ 12^3y dw, 

wherein Ti is that part of F which forms part of the boundary of Gi . Since 
12 = on r, 

/ 12^^3 dw = / I2F3 dx dzdt - / ^H^y dw. 

*^Gx ^4, 'JG-, 

Since dx dz dt is the projection of an element of surface of on the x, z, t space, 
it may be replaced by X0j, ds where ds is an element of 0. If we apply the process 
just indicated to each term of Jg, grad Q x H dw we obtain that 

(15) / grad Q x H dw = / 12 grad (f) x BX ds — 12 cm^l H dw. 

JGi J(t> '■'Gi 

Similarly, for values of H within and on the boundary of G2 , 

(16) / grad Q X B. dw = / 12 grad X H(-X) ds - Q curl H dw. 



MORRIS KLINE (S239) 239 

Likewise, using integration by parts with respect to the variable t, we 
obtain, that 

(17) - / ^tCdw = -\ nCMt ds -\- I nCt dw. 
Similarly, 

(18) - / QtCdw = - I nC{-X)(f)t ds -{- j fiC, dw. 

We now add the left sides and right sides of equations (15), (16), (17)^ 
(18). Equations (13) and (12) tell us that the left side of the sum is zero. On 
the right side of the sum the two integrals over Gy add up to zero because E 
and H are required to satisfy Maxwell's equations in Gi , and for the same 
reason the two integrals over G2 add to zero. If we now combine the remaining 
integrals over </> using the symbol [ ] to denote the jimip discontinuity of the 
enclosed quantity over = 0, we obtain 

(19) [ 12(grad X [H] - <^JC])X ds = 0. 
Similarly we obtain from equation (12b) that 

(20) [ fi(grad X [E] + - 0,[mH])X ds = 0. 

Since these integrals must be zero for any choice of 12 (which vanishes on 
the boundary and outside of G) the other factor in each integrand must be 
zero at each point of 0. We obtain therefore 

(21) grad X [H] - i ^.[eE + F] - 0, 

(22) grad x [^] + ^ 0JmH] = 0, 

which must hold on the discontinuity surface = 0." 

[We shall now use conditions (21) and (22) to dispose of a point left un- 



^^These same discontinuity conditions are arrived at by a different method in R. K. Lune- 
berg, Mathematical Theory of Optics, Brown University, 1944, p. 21. They yield as special cases 
the usual discontinuity conditions on E and H which hold on the boundary \}/{x, y, z) = sepa- 
rating media of different electromagnetic properties. Also, since = ^ = is a discontinuity 
surface, (21) yields E(0+) = -g/(0+). But for the pulse solution /(0+) = 1. Hence 
Eo (0+) = -g/e. From equation (22) we obtain H(0+) = 0. 



240 (S240) ELECTROMAGNETIC WAVES 

settled in section 2 (cf . footnote 2) . As pointed out there, we know from Max- 
well's equations that 

(a) I- (div eE + div F) = 0. 
ot 

The hypersurface = i = is a discontinuity surface for E, H, and F. For 
this (f), grad = and 0« ^ 0. Hence equation (21) tells us that [eE + F] = 
at ^ = 0. Since E and F are f or i < we know that eE + F = for ^ = 0+. 
Since e, E, and F are sectionally continuous, eE = — F for some ^interval about 
t = 0+ and in this ^interval div eE = —div F, that is, no additional /(a:, y, z) 
is needed to integrate (a) with respect to t. 

We now prove that div eE = —div F for all t. Let i^ be a value of t at 
which E or F is discontinuous and suppose div eE = —div F for ^ < ^i . For 
i > ix , the relation div eE = — div F need not hold for a jump in the value of 
either eE or F may introduce an /(x, y, z). However, taking the divergence 
of (21) and using vector identities gives 



(b) -grad</>-curl [H] - ^ div [eE + F] - [eE + F]-grad^ = 



Since jNIaxwell's equation (la) holds on either side of ^ = ^i we may ^\Tite 

(c) curl [H] - i [eE, + F,] = 0. 
We substitute for curl [H] in (b) and obtain 

(d) -grad0-- [eE, + F,] - ^ div [eE + F] - [eE + FJ-grad- = 0. 

c c ^ 

or 

(e; -|- (grad 0-- [eE + F]) - ^ div [eE + F] = 

dt \ C I C 

If we dot (21) by grad we see that the first term in (e) is zero. Now if = 
is a hypersurface for which 0^ ?^ at i = ^i , then [div eE + div F] = on = 
which means again, as in the case = ^ = 0, that no /(a;, ?/, z) can arise by in- 
tegrating (a) with respect to i.] 

Returning to the discontinuity conditions (21) and (22), we see that if e, ju 
and F are continuous at a point (x, 2/, 2;, on = and if = can be re- 
presented in the form i/' — ci = then equations (21) and (22) specialize to 

(23) grad V^ x [H] + e[E] = 0, 

(24) grad i^ x [E] - ^[H] = 
on the surface xj/ — ct = 0. 



MORRIS KLINE (S241) 241 

Equations (23) and (24) are six homogeneous equations in the six quan- 
tities, namely, the components of [E] and [H]. Since discontinuities are assumed 
to exist on \}/ — ct = 0, [E] and [H] are not both zero. Hence the determinant 
of the coefficients must vanish. This condition is both necessary and sufficient. 
However, it is also possible to show^^ that a necessary and sufficient condition 
for non-zero solutions of the six equations is 

(25) rl^l + £ + iPl = efjL. 

This equation is the eiconal equation of geometrical optics and it means that 
^ — c^ = is a family of wave-fronts. Thus at a point in x, y, z, t space where 
e, fi, and F are continuous, but E and H discontinuous, a discontinuity surface 
must be a wave front. 

We shall now derive conditions which the discontinuities of E< and H^ 
must satisfy on a discontinuity surface = 0. Since the Q which was used to 
derive equations (12) is arbitrary (except for the condition on the boundary 
and exterior of G) we may replace it by 0< which must also vanish on the bound- 
ary and exterior of G. We may therefore state instead of (12a) 



(26) 



/ (grad 12, X H - 12,, C) dw = 0, 



We shall now use the fact of equation (13), namely, that the integral over 
G is equal to the integral over Gi plus the integral over G2 . We consider the 
integral over Gi and apply integration by parts with respect to t^ obtaining 

/ (grad fi, X H - 12,, C) dw = / (grad 12 x H - 12,C) dx dy dz 
(27) 

12 X H, - 12,0 dw. 



- / (grad 



Because 12 and 12, are zero on r = Ti + Tg , the ' 'surface" integral over Ti 
vanishes. However, the integral over the portion of </> = contained in G need 
not vanish. 

We now consider the term 

(28) [ grad Q x H dx dy dz. 

In 12 and H we must understand that the variable t is replaced by yp{x, y, z)/c 
which is the form of = when solved for t. However, the gradient operator 



^2Replace [H] in (23) by its value from (24). If E and H are continuous on \p — ct = 
then the condition that \f/ satisfy the eiconal equation need not hold. In this case the condition 
would follow from the next set of discontinuity conditions, namely, equations (39a) and (39b), 
if we use the fact that the right sides are zero for continuous E and H. 



242 (S242) ELECTROMAGNETIC WAVES 

applied to Q applies only to the x, y, z of ^(x, y, z, t). Let us denote by d^Jdy 
the derivative of Q,{x, y, z, yp{x, y, z)lc)^ whereas fl„ denotes the partial derivative 
of fl(rc, y, z, t). Then 

(29) f = «;+io,^„. 

For a typical term of (28), such as J^ 12^,^3 dx dy dz, we may say 

(30) [ n,H, dxdydz = f ^ Hs dx dy dz - - f n^x^.H, dx dy dz. 

<j> ♦/0 oy c J^ 

If we integrate by parts the first term on the right %vith respect to y we obtain 

(31) f ^H, dxdydz = f nH, dxdz - [ n^ dx dy dz. 

''</. oy J Boundary of ^ -U Oy 

In the first integral on the right side we must substitute in 9. and H^ the value 
of y on the boundary of the portion of which lies in G. Since fi is on this 
boundary this integral vanishes. 

As to the second term on the right side of (30) , a theorem on differentiation 
of an implicit function, applied to 4){x, y, z, t) = 0, enables us to say that 

(32) i j Q,xP,Hs dx dydz = j Q.Hsy^j dx dy dz. 

Hence using the facts of (31) and (32) in (30) and remembering that (30) is 
merely a typical term of (28) we may say that 



(33) 



/ grad ^ xK dx dy dz = — j 12 curl H(.t, y, z, - \p{x, y, z)) dx dy dz 

. f _ g rad </) X H -,-,■, 
+ 12, ^ f dx dy dz. 



In (33) the gradient operator applies only to the x, y, z which appear before t 
is replaced by \p/c. However, the curl operator appHes to all the x, y, z. 

We may replace the volume element dx dy dz by X0t ds and (33) becomes 

/ grad 12 x HX</», ds 
(34) 

= - / 12 curl nfa:, y, z, -)\4>t ds -\- \ 9, grad X HX ds. 

We now return to (27) and replace dx dy dz there by Mt ds. We therefore 
have from (34) and (27) 



/. 



MORRIS KLINE (S243) 243 

(grad Qt XU - n.tC) dw 



(35) = I {-Q cui'l H0, + fi, grad x H - 12,0tC jX rfs 

- / (grad Q xHt - 12,C0 (^^. 

We may now apply steps (15) and (17) to the volume integral on the right 
side of (35) except that H, and C^ here replace H and C there. We obtain that 



L 



(grad fi, X H - fi,,C) dw 



(36) 



= {-n curl H0, + ^i grad X H - O^c^.CjX i^s 

- / {12 grad X H, - fi^^C, jX c?s + / (12 curl H, - QCu) dw. 



Of course the same statement can be made for G2 the only change being 
that the normal has the opposite direction over 0. Also if we take the time 
derivative of Maxwell's equation (la), multiply by 12, and integrate over Gi , 
we see that the volume integral on the right side of (36) is zero because one 
factor of the integrand is zero in Gi . Likewise the same integral over G2 is zero. 
We now add (36) to the same equation applied to G2 . We use the remarks just 
made and (26). Then 



j ( - 12 curl nf X, y, z, -j L + 12, grad X [H] - 12,[C]0, JX ds 



(37) 



+ 



/ (12 grad X [H,] - 12[C,]0,)X ds. 



If we multiply (21) by 12^ we see that the second and third terms on the left 
side of (37) vanish. 

Again we have an integral over with 12 arbitrary. Hence the other factor 
of the integrand must vanish and we obtain 



(38a) grad0 x [HJ - ^ [eE, + F,] = -(t>t curl 



H 



(x,^,.,f)j. 



^3 We note that at the discontinuity surface <^ = i = 0, (38a) reduces to 
-(e/c)^,(0+) - (l/c)F,(0+) = -curl H{x, y, z, 0+). We showed in footnote 11 that 
//(0+) = 0. Using the continuity of the derivatives of H in and on the boundary (from one 
side of the discontinuity surface) of Gi , or G2 , we conclude that curl H{0-{-) = 0. For the 
pulse solution Ft{0-\-) = and hence -E'ot(0+) = 0. We may argue similarly from (38b) to 
conclude that Hot{0-\-) = except at points (x, y, z, 0) where charges exist. 



244 (S244) ELECTROMAGNETIC WAVES 

Like^^dse we obtain from equation (12b), by steps analogous to steps (26) to 
(38a), 



(38b) 



grad X [EJ -f " 0.[mH,] = -0, curl \e[x, y, z, ^j\. 



If e, iJi, and F are continuous at x, y, z, t on (j) = and if may be written 
in the form \J/ — d, then 

(39a) grad xP X [H,] + 6[EJ = c curl [H{x, y, z, He)], 

(39b) grad ^A X [E,] - /x[H,] = c curl [E(x, y, z, xP/c)]. 

Relations (38) and (39) hold on the discontinuity surface = and \p = ct, 
respectively/* 

The process used to derive equations (38), may be apphed once more to 
obtain conditions on [E^ J and [Utt]- One starts with the equations (12) wherein 
since fi is arbitrary we may replace it by 12,, . We use the fact that the integral 
over G equals the sum of the integral over Gi and the integral over G2 . Con- 
sider first the integral over Gi , namely, 

/ Gi(grad Q^t X H - ^^^C) dw. 

We integrate with respect to t and obtain (as in step (27)) 

/ (grad 12, X H - 12,, C) dx dy dz - \ (grad 12, x H, - 12,, C,) dw. 

We may now operate on the first integral as in the steps from (27) to (35). In 
fact the only change is that 12< replaces 12 and 12,, replaces 12, . This gives, in 
view of (35), 

/ (- 12, curl H0, + 12,, grad x H - 12,tC<^t)X ds 
- j (grad 12, X H, - 12,, C,) dw. 



i^Conditions (38) and (39) are non-homogeneous while conditions (21) to (24) are homo- 
geneous. This result seems surprising since E, and H, are solutions of Maxwell's equations 
just as E and H are; moreover, E, and H, satisfy definite initial conditions as do E and H. 
However, we have supposed that E and H, if discontinuous, are limits of continuous solutions 
and this condition need not be satisfied by E, and H, . In special cases it could however be that 
E, and H, are limits of continuous solutions, in which cases conditions (38) and (39) would be 
homogeneous. But the discontinuity conditions on higher time derivatives of E and H may 
then still be non-homogeneous. 



MORRIS KLINE (S245) 245 

In view of (21) we shall be able to throw out the last two terms of the 
first integral after we have combined results for Gi and G^ . We therefore neglect 
them now. We next treat the volume integral. Again apply the steps analogous 
to those from (27) to (35) except that H« replaces H and C, replaces C. This 
operation gives 

/ — 12, curl HX</>t ds — \ (—12 curl H,<^, + 12, grad X H, — 12,C,0,)X ds 

+ / (grad 12 X H,, - 12,C,,) dw. 

In view of (38a) we see that we shall be able to throw out those terms con- 
taining 12, after we have combined results for Gx and G2 . We therefore do so 
now and are left with 

/ 12 curl H,0,X ds + / (grad 12 X H,, — 12,C) dw. 

We now operate on the volimie integral as in the steps (15) and (17) and obtain 

/ 12 curl H,0A ds -{- / (12 grad </> x H,, — 12C,,(^<)X ds 

- \ (12 curl H,, - nCtti)dw. 

Again we may throw out the volume integral since it follows from Maxwell's 
equation (la) that in a domain Gi , in which H,, and Cttt exist, the volume 
integral is zero. The same result applies to G2 and adding as in step (19) or 
(37) we get the next discontinuity condition 

grad(/> X [H,J - -(A.[eE,, + F,J = -4>t curl I H,(^a:, y, z, ^jj. 

Similarly, we obtain 

"EJ^x, y, z, -j . 

Again ii cf) = \f/ — ct and if e, ju, and F are continuous onxf/ — d then 



grade/) X [Eu] + - (j)t[^l'Rtt] = -4>t curl 



(40a) grad ^p x [H,J + 6[E,,] = c curl [H,]; 

(40b) grad ^P X [E, J - /x[H,J = c curl [EJ. 

In these last two sets of formulas the curl operation acts on all x, y, z in 



246 (S246) ELECTROMAGNETIC WAVES 

Ht(.T, y, Zy yp/c) and 'Et{x, y, z, yp/c) but the gradient acts only on the x, y, z in 
which are present before substituting yp/c for t. 
We now change notation so that 

A, = r^l and B. = [^1 for . > 

and A_i = B_i = 0. The process we used to obtain equations (23) and (24), 
(39), and (40) obviously generahzes and we may ^mte for A, and B, and v > 

(41a) grad i/' x B, + ek, = c curl B,_^{x, y, z, ip/c), 

(41b) grad \p x A, — juB„ = c curl A,_i(x, y, z, \p/c). 

Relations (41) hold on the discontinuity surface \f/ = ct.^'' 

We have six linear equations for the components of A^ and B, , the verj- 
quantities needed for the coefficients of the asymptotic expansion of the steady 
state fields. Unfortunately the determinant of the coefficients of these linear 
equations is zero, a fact noted in connection with equations (23) and (24). 
Since non-zero solutions of equations (41) are kno^^n to exist yd. the case of a 
pulse source say, it follows that the right sides of (41) must satisfy some addi- 
tional conditions. ^^ We proceed to derive these conditions. 



i^By starting wdth Maxwell's equation and by assuming that the time beha\-ior of the 
solutions is exp { —icot] so that E = u exp { — ico^ } , H = v exp { —icot } , one obtains the time-free 
form of Maxwell's equations involving the functions u and v. If we now assume that 

u=I:Tt4)V- and v=i:v4)V- 

substitutes in Maxwell's equations, and equates to zero the coefficients of like powers of l/ik, 
we obtain conditions (41). (See Luneberg, Mathematical Theory of Optics, Brown University 
Notes, 1944, Volume I, p. 81.) This process assumes the existence and differentiability of the 
asymptotic series for u and v. (Existence is established in this paper at least for a class of 
solutions of Maxwell's equations.) Further, the process just indicated gives no inkling of the 
relationship of the U^ and V^ to the discontinuities of the pulse solution Eo , Ho . One could by 
an indirect argument which relies upon the uniqueness of asymptotic expansions subsequently 
identify the U„ and V^, with these discontinuities. However, were the substitute procedure 
presented carefully it might at best be slightly briefer at the expense of directness and insight. 
Incidentally, conditions (41) as given by Luneberg have a minus sign on the right side but this 
difference is due to the difference in signs between the asymptotic expansion (6) of this paper 
and the expansion assumed by Luneberg in the Brown notes. 

i®At first sight one might expect to solve equations (41) by a purely algebraic procedure, 
since these are just six linear equations in the six components of A^ and B^ . However, it can be 
shown that the rank of the matrix of the coefficients is four. Hence even if one proceeded to 
solve, relying upon the fact that non-zero solutions are known to exist, the solution can at best 
express four of the components in terms of the other two. The solutions obtained for any one 
value of V must be chosen so that, when used on the right side of equations (41), it would be 
possible to solve for the A^+i and B,+i ; in other words, the A, and B, must be chosen so that 



MORRIS KLINE (S247) 247 

From equation (41a) 

1 c 

A, = — grad i/' x B, + - curl B,_i . 

We substitute this quantity in (41b) and obtain 

— grad lA X ( - grad \l/ X B J -1- grad xp X - curl B,_i — fiB, = c curl A,_i . 

Applying the vector identity for the vector triple product gives 



(43) 



— - (grad rp-B,) grad \p -\- - (grad ^pfB, — juB, 



- grad \p X curl B,_i + c curl A,_i . 



We form the scalar product of both sides of this equation with Ao and use the 
fact ^^ that Ao • grad xp = 0. We obtain 

1 c 

- (grad \l/)Ao'B, — Aq-B^ = — - Ao-grad \p X curl B,_i + cAo-curl A,_i . 

By equation (24) Ao X grad xp = —(jBo . Hence, first interchanging dot and 
cross on the right side, we obtain 

1 c 

-(grad \pyAo-B, — /iAq-B, = -f- /xBo-curl B,_i + cAq • curl A,_i . 

In view of equation (25) we obtain 

(44a) jLtBo-curl B^_] + eAo-curl A^_i = 0. 

If we form the dot product of both sides of (43) with Bo and use the fact^^ 
that Bo • grad \J/ = we get 

^i (grad xPYB, - mB.J-Bo = -^ Bo -grad i^ x curl B,_i + cBo-curl A,.i . 



the next set of equations in the recursive system (41) is consistent. But these consistency con- 
ditions are precisely the equations (45) which we derive here and unfortunately these conditions 
are partial differential equations. Hence we are led, as in the procedure of the paper, to solve a 
mixed system of algebraic and partial differential equations. 

^^This fact follows at once from equation (23) if we form the scalar product of grad ^i' with 
the left side. Likewise we get from (24) that Bo -grad \p = 0. 



248 (S248) ELECTROMAGNETIC WAVES 

If we interchange dot and cross and use (23) and (25) we get 

(44b) Ao- curl B,_i - Bo • curl A,_i = 0. 

Equations (44) are the conditions which hold for the right sides of equations 
(41) in view of the fact that non-zero solutions must exist for A^ and B, . Since 
equations (41) hold for each value of ^ > 0, we can state equations (44) for v 
instead of i' — 1, namely, 

juBo'Curl B^ + eAo- curl A, = 0, 
(45) V > 0. 

Ao-curlB, — Bo -curl A;, = 0; 

Equations (41) supplemented by equations (45) are the discontinuity condi- 
tions which relate solutions of Maxwell's differential equations on the two sides 
of a discontinuity surface \l/ — ct = 0. It is evident that these conditions are 
consequences of the integral formulation of Maxwell's equations. 

5. The Ordinary Differential Equations for the Discontinuities A, and B, 

We shall use equations (41) and (45) to determine A^ and B^ . ^liile 
equations (41) are just ordinary algebraic equations, equations (45) are first 
order partial differential equations. Instead of attempting to solve these partial 
differential equations directly we shall convert them, vdth. the aid of equations 
(41), to a system of recursive ordinary differential equations. The process 
which will be used generalizes the one employed^^ to obtain the ordinaiy differ- 
ential equations for Aq and Bo which hold along the rays, that is, the cur^^es 
which are orthogonal to the wave fronts xp — ct = 0. The latter differential 
equations are known as the transport equations of geometrical optics. The 
additional ordinary differential equations which we shall obtaiQ and which t^tU 
give the variation of A^ and B^ along the rays may properly be called the higher 
transport equations. 

Because the derivation of the ordinary differential equations is lengthy it 
has been treated separately in the appendix. The results^^ are: 



i^See Luneberg, Brown Notes, Article 11, pp. 41-46. See in particular equations (11.38) 
of this reference. 

^^Equations (46a) and (46b) here are equations (48) and (49) of the Appendix. It was 
remarked in footnote 15 that the discontinuity conditions (41) can be derived by assuming the 
existence and differentiability of an asymptotic solution to Maxwell's equations and by substi- 
tuting in the equations. It is worthy of note that equations (46) can be obtamed formally by 
substitution of the asymptotic solution in the second order equations for E and H derived from 
Maxwell's equations. This fact has been noted by H. J. Riblet in an unpublished paper and 
by F. G. Friedlander in ''Geometrical Optics and Maxwell's Equations," Proc. Cambridge 
Phil Soc, Vol. 43, Part 2, April, 1947, pp. 284-286. The comments made in footnote 15 
a propos of the procedure discussed there apply to the procedure mentioned here. 



MORRIS KLINE (S249) 249 



(46a) ^ ^ + ^^^'^^ + n *^^'"^^ ^'^"'^ ^""^^ ^ ^ "C. 



rTR 2 

(46b) 2 V + ^^^^^ + - (g^ad n-B.) grad ^p = -D, , 

wherein 

C„ = M curl (- curl A^_i ) — grad (-div eA^_i j, 

D^ = e curl (- curl B;,_i I — grad (-div /xB^_i 1, 
Co = Do = 0, 



It is further understood that equations (46) represent differentiation along the 
rays orthogonal to the wave fronts yp — d = Q^r being any convenient parameter 
in terms of which the equations of the rays can be expressed. 

To use equations (46) we note first of all that they are a series of recursive 
vector differential equations for the six quantities, the components of A^ and 
Bp . Thus Aq and Bo must be determined first and then used on the right side 
to solve the next case for j' = 1, etc. 

We note next that one needs the quantity yp{x, y, z) to fix the coefficients 
in (46). Now \p satisfies the eiconal equation -tpl -\- xf/l -\- ypl = efi and hence 
one must solve this partial differential equation. In the simple case of a dipole 
in a non-homogeneous medium there will be a single set of expanding wave 
fronts. If, however, there are obstacles or other discontinuities in the medium 
there will be wave-fronts reflected from the discontinuities. Not only must 
these wave fronts be determined but the differential equations (46) must be 
solved along the rays to these wave fronts also. The existence of more than 
one set of wave fronts means that at any point x, y, z of space more than one 
discontinuity surface \l/(x, y, z) = ct will pass and the several values of A, and 
B;, at this point will be required to obtain the asymptotic expansion (6). That 
is, a summation over a will have to be applied, the various A^ and B, at one 
point being actually then the several values which must be sununed. 

A third point to note in regard to the use of equations (46) is that we 
want the known coefficients to be expressed as functions of r, the parameter 
along the rays. This means that the quantities A^^xf/, grad n, n, and grad \l/ 



250 (S250) ELECTROMAGNETIC WAVES 

must be expressed as functions of r. Theoretically this can be done for, from 
a knowledge of the wave fronts one can obtain the differential equations for 
the rays and then solve these equations for the rays.^^ Once we obtain the 
equations of the wave fronts and the rays we can at least fix the coefficients 
on the left side of (46). We must still determine the quantities on the right 
side before solving for A^ and B^ . Now the quantities C, and D, will be known 
to us as functions of r from the solution of the preceding differential equation 
in the recursive set. In order to determine such quantities as curl A,_i and 
div A^_i we need A^_i as a function of x, y, z. For this purpose we may invert 
the equations of the rays themselves. To each family of wavefronts there 
corresponds a two parameter family of rays. These two parameters ^Hl be 
involved in our functions A^_i and B;,_i because we substituted the equations 
of the rays into -^^ n, and other quantities in order to determine A,_i and B,_i . 
We may now take the equations of the rays 

(47) X = x{t, a, /3), y = 2/(t, a, /3), s = .s(t, a, /3) 

and solve these for a, ^, r in terms of x, y, and z. Substitution of these values 
of a, ((3, and r in A^_i(r, a, /?) and B;,_i(r, a, (3) converts them into functions of 
x, y, and z. 

The solutions of equations (46) for the case j' = (in which case C, = 
D^ = 0) give the geometrical optics behavior of the field generated by a source 
along any particular ray emanating from that source, for expansions (6) show 
that, except for a phase factor, Aq and Bq give the field generated by a mono- 
chromatic source as the wave length approaches zero.^^ ^Miile the equations 
for Ao and Bo have been derived by other methods it is the asymptotic expansions 
(6) which show clearly what they furnish. 

6. Introduction of Initial or Boundary Conditions for A, and B, 

The discussion of equations (46) in the preceding section ignored the fact 
that solution of these equations introduces arbitrary constants. Hence thus far 
A^ and B^ are not uniquely determined. In fact the differential equations (45) 
and linear equations (41), as well as the resulting ordinarj^ differential equations 
(46), hold for all solutions E and H of the integral form (12) of Maxwell's equa- 
tions at all points x, y, z at which F, e, and fx are continuous. For the purpose 
of determining the coefficients of the asymptotic expansions (6) we \\ish the 
A^ and B^ belonging to the pulse solution and this solution, as well as the other 
ones we are dealing with, is determined by the source. 

In general there will be several (perhaps an infinite number of) fimctions 



^oSee article 10 of Luneberg, Bro^n Notes, Volume I. It is also possible to obtain the 
wave-fronts from a knowledge of the rays. See articles 8 and 9 of this reference. 
2^6'/. p. 84 of Luneberg, Brown Notes, already referred to. 



MORRIS KLINE (S251) 251 

A,(t), B,{t) for a given v. One pair, A, , B, , states the behavior of the discon- 
tinuities of the Mh time derivatives of E and H along the rays belonging to the 
wave-fronts emanating from the source and for the moment we shall confine 
ourselves to this pair. The parameter r in equations (46) can be time itself or 
something directly related to the time. At r = 0, the ray is at the source and 
so knowledge of A, and B, at r = can be expected to depend upon knowledge 
of the source. Indeed it appears possible (compare footnotes 11 and 13) to 
determine A^ and B^ at t = in terms of the source function F and its time 
derivatives. However, for the very important case of a dipole or point source 
the function g(x, y, z) in ¥ = gf{t) is the 5-f unction and it is awkward to work 
with it. 

We shall therefore dispose of the problem of uniquely fixing the A^ and 
B^ of the field directly transmitted by the source in an alternative way which 
is consistent with the preceding remarks and which should give the same result. 
If these quantities were known at one point on a ray they could be fixed uniquely. 
Now the quantities A^ and B, are to be substituted in the expressions for u 
and V given by (6). Hence we may look to the problem for which u and v give 
the solution to find suitable initial or boundary conditions for A^ and B„ . 

If, for example, u and v represent the spatial factors of the steady state 
field of a dipole in a non-homogeneous medium, then we should want as a suit- 
able boundary condition on u and v that they satisfy at the dipole the boundary 
conditions on the dipole when placed in a homogeneous medium. We could 
therefore require that each term of the asymptotic expansion for the u and v 
which represent the dipole field in a non-homogeneous medium have the same 
limiting value at the dipole as does the corresponding term of the asymptotic 
expansion for the dipole in a homogeneous medium. Now the asymptotic 
expansion for the dipole in a homogeneous mediiun can be obtained in other 
ways^^ and is 

u = exp {ikr} H- ,2 exp {ikr] 

T KT 

fAo\ p X (m X e) ( ., ) 

(48) V = ^ — exp {ikr] 



(p X (m X 9)i — (m- 9)922) exp [ikr] (3(m- 9)9 — m) exp [ikr] 
kr' "^ kV 



wherein the vector 9 = (x, y, z)/r and m is a constant complex vector giving 
the moment and orientation of the dipole located at (0, 0, 0). The quantity r 
in exp {ikr} corresponds in a homogeneous medium, to the \l/{x, y, z) of our theory. 
Examination of equations (48) shows that as r approaches each term 
becomes infinite. Hence the limiting values approached by each tenn of (48) 



^^Luneberg, Brown Notes, Volume I, p. 79. 



252 (S252) ELECTROMAGNETIC WAVES 

cannot be used directly as boundary conditions; however, a simple reformulation 
can be. The expansion coefficient of rays in a non-homeogeneous mediimi^^, 
denoted by K, is equal to r^ in a homogeneous medium. If we multiply the 
n-th term in each expansion of (48) by {K^^'^Y^^ (the first term given by n = 0), 
we see that each term approaches a finite limit. The asymptotic expansions of 
u and V for a dipole in a non-homogeneous medium will not contain simple 
inverse powers of r^. These will be replaced by some quantities invohdng K. 
However, to meet the boundary condition suggested above we can require that 
the n-ih. term be multiplied by {K^^^Y^^ and that the limit of this product as it 
approaches the dipole be the limit given by the corresponding term of (48) 
multiplied by r"^\ But otherwise, the limit of (Z^^^)''"^A, and {K^^^Y'^B^ must 
approach the limits of the corresponding terms of (48) multipHed by r'^\ It 
will be noted that except for the first two values of v in the case of u and the 
first three in the case of v, the limits which {K^^^Y^^^v and (Z^'^j'^^B, must 
satisfy are zero. 

This method of determining the arbitrary constants in the expressions for 
A, and B, takes care of that summand of the I'-th coefficient of u and v of (6) 
which corresponds to the V^a of the directly transmitted set of wave-fronts. 
For wave-fronts which arise from other causes, such as the presence of discon^ 
tinuities in € and ju, other initial conditions now being investigated must be used. 

7. Conclusion 

Theoretically the problem of obtaining the successive coefficients of the 
asjrmptotic expansions for u and v, the spatial factors of time harmonic E and 
H, can be solved without knowing the full pulse solution Eq , Ho . By utilizing 
the fact that the coefficients depend only on the discontinuities of Eq , Ho and 
their time derivatives on the wave-fronts passing any point in space, it is possible 
to obtain a system of first order ordinary differential equations whose solutions 
are these coefficients. Solution of even the first two sets of equations of this 
recursive system, that is the cases v = {) and v = 1, would give a significant 
improvement over geometrical optics approximations to time haraionic fields. 

The theory of this paper can be applied at least theoretically to find the 
asymptotic form of the dipole field in a non-homogeneous medium. There is 
little doubt that it can be carried further to treat problems in which more 
general media and boundary conditions beyond that of the dipole are involved. 
However, further work is needed on this extension of the theory. A difficulty 
which at the present writing is not resolved is that one must know all the families 
of wave fronts and rays which arise from the source and boundaries, for contri^ 
butions to each coefficient of the asjonptotic expansion will come from each 
wave-front through a point in space. 



23For the definition of K see Luneberg, Brown Notes, Volume I, p. 50. K is the reciprocal 
of the Gaussian curvature of the wave-front. 



MORRIS KLINE (S253) 253 

Appendix 

Derivation of the Ordinary Differential Equations for A, and B, 

This appendix will derive the ordinary differential equations (46) for 
A, and B, by utilizing equations (41) and (45). Equations (45) are restated as 

(1) Ao • curl B, - Bo • curl A, = 0, 

(2) eAo • curl A, + /xBq • curl B, = 0. 
Equations (41) are 

(3) grad \f/ xB, -{- ek, = c curl B,_i , 

(4) grad yp X K — )uB^ = c curl A,_i . 
Dot both sides of equations (3) and (4) by grad \p. Then 

(5) € grad xp-A^ = c grad i/^curl B^_i , 

(6) M grad xJ/B^ — —c grad i/'-curl A^_i . 

Taking the divergence^* of both sides of (3) and (4), and using the fact 
that div curl = 0, we get 

(7) div ek, = —div (grad \p X B^) = +grad \^-curl B, , 

(8) div juB^ = div (grad ip x k^) = —grad \f/-cuY\ k^ . 
From (5) and (6), 

(9) grad yp-k^ = -grad i/'-curl B;,_] , 



(10) grad yp-B, = — grad i/"curl A,_j . 

Using the first and third members of (7) and (8) iox v = v — 1 and sub- 
stituting in (9) and (10), respectively, gives 



2^In taking the divergence of both sides of (3) and (4) we must recall that the curl operator 
in (3) and (4) applies to A^_i and B^_i as functions of x, y, z and \p{x, y, z). Hence the diver- 
gence is to apply in the same sense to the functions on the left. We therefore understand here- 
after that in all quantities A^ , B^ for all v, t is first replaced by \l/{x, y, z)/c. This is the 
correct sense of equations (1) to (4) for they hold on the discontinuity surface \l/{x, y,z) — ct = 



|.,/ 



254 (S254) ELECTROMAGNETIC WAVES 



(11) gracl ^p'A, = - div €A,_i , 



(12) grad ^p■B, = -divMB,_i 



M 



From (4), 



(13) B. = ^^^^^ X A, - %url A,., . 
We substitute this in (2) and obtain 

(14) eAo- curl A, + mBo- curl (^^^^^ x A,j - mBo- curl f- curl A,_ J = 0. 
Now eAo = —grad i/' x Bq by equation (23) of the text proper. Hence 

Bo X grad i/^- curl A, + ^Bq- curl ( ^^^ x A J 
(15) 

— juBo'Curl (- curl A^^i ) = 0. 

Dividing by ju and interchanging dot and cross give 

Bo .2?^ X curl A. + B„.cui-1 (^^^ x a) 
(16) 

— Bo- curl (- curl A^_i j = 0. 

Now by a vector identity and the fact that curl grad = 0, 

A, X curl f ^^^ ^j = A, X (grad - x grad \pj. 

Dotting both sides by Bq gives 

(17) Bo -(a, X curl ^^-^) = Bo -(a. x (grad - x grad ,a)). 

Now let us add (17) to both sides of (16) and transfer the last term on the 
left side of (16) to the right side. We obtain, after rearranging terms, 

T» / 1 /grad \p . \ , grad \^ i . . a i gi'^d \f/ 
Bo -i curl - — ~ X A J + X curl A, + A, x curl ^^ 



(18) 

= Bo -curl (- curl A,_i j + Bo-^A, x (grad - X grad x// 



MORRIS KLINE (S255) 255 

We shall now use the following vector identity/^ 

curl (A X B) -f A X curl B -\- B x curl A 



In this identity 



= -2 ^ - B div A + A div B -f grad (A-B). 
da 



£ = ^-l; + ^4 + ^'i = ^^-^^ 



where Ai , A2 , and A^ are the components of A; the s^-mbol d/da denotes 
differentiation in the direction of the vector A. In our case A = (1/m) grad 
^ and B = A, . Then 

At = - rp, , A^ = - xp„ , As = - rp, . Let — = lA, — -f v^. T- + ;A. t:^ 
At fi fjL OT ox oy oz 



Then from (18) 

= Bn-curl 



B,.^ -? ^ - A, div (^) + ^^^^^div A, + grad f^^^-A, 



(19) 



f- curl A,_J + Bo-S A, X (grad - X grad xl/JK 

= Bo -curl (- curl A,_i 1 

+ Bo -UA,- grad \p) grad - — ( A, -grad -j grad \p}. 
By a standard vector identity 

grad ( - (grad xp • A,) ) = grad - (grad i^ • A,) + - grad (grad j^ • A,) . 



2^This identity results from adding the standard vector identity for curl ( A X B) and an 
identity for A X curl B + B X curl A. The latter identities are readily proved. 



A X curl B = A X U" X — j + A X U X — ) + A x ( ^' X 



by I ' -- - \- - Q^^ 

aB _ aB 

dyJ'' ' V* dzl ^^' dx ^"" dy ^ ^ dz' 



= (-i>+(-i>+(-f)--^.f --'.---'■ 



Likewise we form B X curl A and add. See Appendix I, p. 7, of Lunebcrg, Brown Notes, 
Volume II. 



256 (S256) ELECTROMAGNETIC WAVES 

If we replace the last tenn on the left side of (19) by the equivalent just indi- 
cated we note that a term on the new left side cancels one on the right. Also 
since Bo • grad yp = we may throw out some terms and obtain 

B„.(-? f^ - A. div (e^) + i grad (grad ^-A,) 
(20) 

= Bq • curl ( - curl Ap_i ) . 

By (11) we have 

B„.{-2^_A.div(E-ii 

{ fi dT \ M 

= +Bo 
Now 



(21) 



curl ( - curl A,_i 1 — - grad ( - div eA,_i 1 f. 



by definition of the S3rmbol A,,\J/. Hence, multiplying by fx and a minus sign in 

(21) gives 

(22) Bo -(2 ^j + {A.Xf) = -Bo-L curl (- curl A„_,j - grad (^div €A,_,)|. 
We may rewrite equation (22) as 

(22a) Bo -12 ^ + A,A,;A + M curl (- curl A,_i) - grad (^div eA,_,)| = 0. 

In so far as dkjdr is concerned, this is a derivative in the direction of grad yp. 
If we regard x, y, z m k^ as functions of r, the parameter along the ray per- 
pendicular to grad \^, we may wTite instead dAJdr. 

We note that the second factor in equation (22a) is a vector perpendicular 
to Bo and hence in the plane of Aq and grad \^. We can go through a series of 
steps analogous to those above to show that this quantity is also perpendicular 
to Ao and so must have the direction of grad \f/. The steps are readily indicated, 
for the changes over what was done above are minor. We need go back only 
to step (13) which gives an expression for B^ and substitute this value of B, in 
(1). Then 

(23) 



o-jcurl ^^^^^ X A.) - cm-l (- curl A,_,)| - Bo-cm'l A, = 0. 



1 



MORRIS KLINE (S257) 257 

From equation (24) of the text proper Bo = (1/m) grad ^ x Aq = — (l//z)Ao 
X grad xp. Hence 

(24) A„.{curl(E:^ X A.) - curl [^ curl A._,)} + A„ x S^-curl A. = 0. 

Interchange dot and cross in last term. If we compare (24) and (16) we see 
that we can repeat everything we did from (16) on to (22) and conclude that 
(22a) holds with Aq replacing Bo . Hence the second factor in equation (22a) 
is a vector perpendicular to both Ao and Bo . Since Ao , Bo , and grad yp are 
mutually perpendicular, this factor is parallel to grad \l/. We may say therefore 
that 



(25) 



2 -^- + A,A^^ + fi curl (- curl A,_i j — grad (-div €A,_i ) = R grad \py 



where /2 is a scalar function to be determined. To save writing we let C, = 
fi curl (c/m curl A^_i) — grad (c/e div eA,_i). Then 

(26) ^ ^ "^ ^'^"^ + 0, = R grad ^. 

C, thus stands for two terms, which will not be affected by what follows. 

We now determine the value of R, Dot both sides of (26) by grad ^. 
This gives 

(27) 2 ^-grad rP + A,M»-grad ^ + C,-grad ^ = Rn\ 

Cut 

Using the fact that n = i^l -\- ^l + ^lY^^ we prove readily, since 

that 

(28) n grad ^ = T~ (grad \f). 
Hence 

(29) -J- (A, -grad \f) = -y-^-grad i/' + A,-n grad n. 

UT UT 

We may therefore rewrite the value of Rn^ in (27) as 

(30) Rn = -2riA,-grad n -}- 2 — (A, -grad rP) + A^\^A,-grad ^ + C,-grad \p. 






258 (S258) ELECTROMAGNETIC WAVES 

Hence 

(31) E = --gradn-A, + P, 

where 

/ortN r> 2i<^ .. J /\ I A / A,-grad r^ , C,-grad ^ 

(^2) ^' = ;? ^ ^^'-^^^ ^) + ^^^ — ;? — + —7? — • 

We shall show that P, = 0. From (3), wherein we dot both sides by grad 
yj/j we obtain 

(33) €A,-grad yp = c curl B,_i-grad i/^. 
Using equation (11) gives 

(34) div €A^_i = curl B,_i-grad yp. 
Since our relations hold for all v > 0, 

(35) div eA, = curl B,-grad yp. 

If we substitute the value of B„ from (4) into (35) we get 

(36) div eA, = curl (^ x A J -grad yp — curl (- cur) A,_i j-grad yp. 

Now let 

[e^, aJ = curl (e^ X A,) 
(37) 

+ S2^XcurlA, + A„xcurie5^. 

Dot both sides of (37) by grad yp and use (36). Then 

div eA. = [e^, A.]-grad ^ - (a. X cui-1 S:^) • grad ^ 

— curl (- curl A^_i j-grad yp. 



(38) 



Using a vector identity for the curl of a scalar and a vector in the second term 
on the right side gives 

did eA, = — -, A, -grad yp — [k, X (grad - X grad ;/'jj-grad yp 

(39) 

— curl (- curl A,_i j-grad yp. 



MORRIS KLINE (S259) 259 

Apply the vector identity for the vector triple product to the second term on 
the right side. We obtain 

div eAy = -, A, -grad \p — (A,-grad i^jlgrad --grad ypj 



(40) 



+ 



(a, -grad -\n — curl (- curl A,_i j-grad yp. 



Now the quantity [(grad tA)/m, A^] given by (37) is precisely the quantity 
encountered in (18) and converted by a vector identity into the form in (19). 
If we make the same change in (40), expand grad [(grad i/')/)u-A,] into 
grad 1//X (grad i/" AJ + (l//i) grad (grad ^-K), and cancel terms we get 

div cA^ = — - -r-^-grad yf/ — (A, -grad \l/) div [- ~] 

II a T \ 11 / 

(41) + e div A, + - grad (A, • grad \}/) • grad ^ 



+ n^( A,-grad -) — curl (- curl A^_i j-grad yp. 



By a vector identity, div eA, = A, -grad € + € div A, . If we use this 
equation in (41) and multiply by /x, we obtain 

= -2 ^^grad ^ - M(A,-grad ^) div (^^^ 

(42) + grad (A^-grad i/')-grad ^ — /zA^-grad e 

+ ju^^( A,-grad -j — n curl (- curl A^_i j-grad \p. 

We now use the fact noted above in connection with (22), namely, that 

M div [(grad yp)/n)] = A^-p. Also, since 

the quantity grad /-grad xj/, where/ is any scalar, is df/dr. Hence 

(43) grad (A, • grad xp) - grad \f/ = — {A„- grad \p) . 

UT 

The use of these several facts in (42) gives 



260 (S260) 

= 



ELECTROMAGNETIC WAVES 



dA 



2 -7-^-grad ^ — A^)/'(A,-grad ^) 

UT 



(44) + ^ (A,-grad if) - iiA^-grad e 

+ /x^^( A^-grad -j — ju curl (- curl A,_i 1-grad ^. 
If we use (29) to eliminate the first term in (44) we obtain 
= —-J- (A^-grad rp) + 2A,-n grad n 

UT 

(45) — A^^(A,-grad \f/) — )uA,-grad e 

— €(A,-grad n) — ii curl (- curl A,_i j-grad \f/. 

Because 2n grad n = grad ri = grad en = e grad m + m grad €, the second, 
fourth and fifth terms cancel and we have 

(46) = 4--T- (A, -grad \p) + A^\f/{A, • grsid id -\- n curl (- curl A,_i j-grad \^. 
From (32) and the value of C, it follows that 

(47) n^P, = 2 — (A, -grad \//) + A^\l/{A, - grad \f) + fi curl (- curl A,_i j -grad ^ 

— grad (-div (eA^.j) )-grad x//. 

By (11) and (43) we may replace the last term on the right by (A, -grad \l/) d/dr. 
Hence, comparing (46) and (47), we see that P, is zero. 

We note this fact in (31) and then substitute the value of R from (31) in 
(26). We have finally 



(48) 



2 ^ + A,A,^ + - (grad n- AJ grad rp = -C. 



where C, = fx curl ((c//x) curl A,_i) — grad {(c/e) div eA^-i). 

To obtain the analogous ordinary differential equation for B, we note first 
that steps (1) to (12) treat both quantities and are basic to (48) as well as the eq- 
uation we now seek. Starting then with a step analogous to (13), this time we 
substitute the value of A^, from (3) into (1) and obtain 

(140 Ao- curl B, - Bo-|curl (-- grad ^ x B J + curl f- curl B,_ J | = 0. 



(150 



(160 



MORRIS KLINE (S261) 261 

Again we replace Ao by — 1/e grad yp xBq and obtain 

— grad ^ X Bo • curl B, 

- Bo-Scurl (-- grad i^ X B J + curl f- curl B,-ij| = 0. 

Bo.^^^^^XcurlB,+ 

Bo- curl (^^^ X B.) - Bo- curl (^ curl B,_i) = 0. 

Analogously to the derivation of (17) we derive 

(170 Bo-(b. X curl ^^^) = Bo-|b, X (grad ^ X grad ia)|. 

We add both sides of (170 to (160, transfer the last term of (160 to the 
right side and obtain 

Bo -{curl (e?^ X B,) + a^ X curl B. + B, x curl ^} 

= Bo- curl f- curl B,_i j + Bq-SB, x (grad - X grad xp)?. 
Again we use the vector identity as in the step from (18) to (19) and obtain 
B„.{-? f^ - B, div (S^) + eSpdiv B, + grad (^-B,)} 

= Bo- curl f- curl B,_ij + Bo-MB, x (grad - x grad \pW, 

= Bo- curl ( - curl B,_i 1 + jSo-uB^-grad ^) grad - 

- |^B,-grad-j grad rp?. 
We now use the vector identity 

grad \- (grad ^^-B^,)^ = grad - (grad ^-BJ + - grad (grad ^-B,) 
to replace the last term on the left side of (190 by its equivalent, cancel terms, 



(180 



(190 



262 (S262) ELECTROMAGNETIC WAVES 

use the fact that Bo-grad i^ = to throw out the third term on the left side 
and the last on the right, and obtain 

(200 

= Bo-curl (- curl B,_, ). 

By (12) we have 

B..{-ff-B..i,(«=P)} 

(210 

= Bo -^ curl (- curl B,_i) — - grad (- div /zB,_i 1^. 

Again using div (grad i/'/e) = (1/e) A.r/' and multiplying by e we obtain 

(22') Bo-(-2 ^ - B,A,;a) = Bo'je curl (^ curl B._,) - grad (^div mB,_x)|. 

By carrying the analogous argument thus far we see what the equation for 
B, will look like. It is clear from comparing (22) and (22') that the ordinary' 
differential equation for B^, reads 

(49) 2f + B.A.^ + 2(af^.B,).grad ^ = -D. , 

where D, = e curl (c/e) curl B^_i — grad ((c/m) div mB._i). 
Of course the argument required to establish (49) from (22') must be carried 
through but the only question, after noting that e replaces /x and )u replaces e 
in (22') as against (22), is a possible difficultj^ with signs since equations (1) 
and (2), (3) and (4), and (23) and (24) of the text are asjTimietric with respect 
to signs. However these differences offset each other and (49) is the correct 
equation for B, . We understand that for »' = 0, C, = D, = 0, for equations 
(3) and (4) in this case have zero on the right side. 



Field Representations in Spherically 
Stratified Regions* 



By NATHAN MARCUVITZ 
Polytechnic Institute of Brooklyn 



1. Introduction 



The systematic treatment of electromagnetic radiation and diffraction 
problems in spherically stratified regions requires the ability to obtain a repre- 
sentation of the vector electromagnetic field produced by prescribed and induced 
sources in the given region. Such representations may be obtained by a sys- 
tematization of the classical method of separation of variables termed variously 
the method of characteristic (or eigen) functions or the method of guided 
waves (or modes). The desired field representation is expressed as a super- 
position of mode functions that are so chosen as to permit a simple evaluation 
of the associated amplitude functions. The problem of finding such a set of 
modes leads to one or more eigenvalue problems of the Sturm-Liouville type in 
which the eigenfunctions are characteristic of one or more of the spherical (r, 6, <p) 
components of the wave operator — V^. These mode functions form a com- 
plete set of orthogonal functions in either the r, ^ or <;^ directions and possess in 
general both a discrete and continuous spectrum. With the determination of 
such a set of mode functions the original three dimensional field problem may 
be reduced to a one dimensional problem for the mode amplitudes. The latter 
is an ordinary differential equation problem characteristic of wave propagation 
along a single direction — the transmission direction, and is phrased advan- 
tageously as a generalized transmission line problem. 

The indicated reduction constitutes a '^diagonalization" procediu^e that can 
be effected in various ways depending on whether one employs a representation 
in terms of eigenf unction of d and (p, i.e. waves guided along r, or, if there is 
no <p dependence, in terms of eigenfunctions in r, waves guided along 6, etc. 
Each of the resulting field representations has a rate of convergence dependent 

Paper presented at the June, 1950, Symposium on the Theory of Electromagnetic Waves, under 
the sponsorship of the Washington Square College of Arts and Science and the Institute for 
Mathematics and iVIechanics of New York University and the Geophysical Research Direc- 
torate of the Air Force Cambridge Research Laboratories. 

*This work was performed at Washington Square College of Arts and Science, New York 
University and was supported in part by Contract No. AF-19(122)-42 with the U.S. Air Force 
through sponsorship of the Geophysical Research Directorate of the Air Force Cambridge 
Research Laboratories. 

263 (S263) 



264 (S264) ELECTROMAGNETIC WAVES 

upon the parameters involved. In the problem of radiation from an antenna 
above a spherical earth with various atmospheric conditions, the representation 
in terms of guided waves along 6 is usually the most convergent. All repre- 
sentations are interrelated, however. Starting from a representation with poor 
convergence, one can obtain by a summation technique the representation with 
better convergence. It is this latter procedure that has been employed by 
Watson, van der Pol, Bremmer [1], et al. in their discussion of spherical earth 
problems. The direct application for such problems of the rapidly convergent 
representation in waves guided along 6 has been made by Booker and Walkin- 
shaw.^ Although such applications are considered below, our ultimate interest 
lies in the general representation theory necessary for the solution, via the 
theory of guided waves, of diffraction problems involving discontinuities in 
spherical regions. 

The general electromagnetic diffraction problem involves the solution of 
the vector field equations with arbitrary electric and magnetic current sources. 
The sources are either prescribed or induced (and hence initially unknown), the 
latter arising from the presence of discontinuities. In the steady state for 
which a time dependence exp {+jcot} (or exp [—icot]) is suppressed, the rms 
electric field E and the rms magnetic field H are determined by 

V X E = -jkfiK - M 
(1.1) 

V X H = iA:€ E + J 

where M(r) and J(r) are respectively the magnetic and electric current densities, 
/x(r) and e(r) (functions of r only) are the relative dielectric constant and the 
relative permeability, and k is the wave number in vacuum. For mathematical 
simplicity the normalization of the field quantities has been so chosen that the 
intrinsic impedance (mo/co)' of vacuum is unity. It is desired to obtain in terms 
of J and M a solution to equation (1.1) satisfying such boundary conditions as 

(1.2) nX E = = H-n 

on perfect metals, where n is the normal to the metal surface; or such conditions 
as finiteness and single- valueness in closed but unbounded regions, etc. 

In Section 2 the reformulation of the vector problem posed by equations 
(1.1-2) as ordinary scalar problems of Sturm-Liouville type is considered in 
detail. There is presented herein the basic mode representation theory necessary 
for the solution of special and general field problems in spherically stratified 
regions in terms of guided waves along either the r or 6 directions. The de- 
termination of the field of an arbitrary current source is reduced hereby to the 
solution, as a function of the mode index, of an ordinary second order inhomo- 
geneous differential equation, or equivalently, of two simultaneous first order 

i^Mode Theory of Tropospheric Refraction . . . ." Joint Conference of Phys. Soc. and 
Royal Meter. Soc, April, 1946, Appendix II. Also H. Bremmer, loc. cit. p. 202-7 and A. 
Sommerfeld, Partial Differential Equations, Academic Press, New York, 1948, p. 214. 



NATHAN MARCUVITZ (S265) 265 

(transmission line) equations. In Section 3 a general 8 function procedure for 
the explicit solution of the eigenvalue problems formulated in Section 2 is 
discussed via the methods of Weyl, Titchmarsh, et al. The interrelation of 
eigenvalue and transmission line problems in terms of the Green's functions 
that characterize both types of problem is pointed out. In Section 4 complete 
sets of mode functions of use in typical spherical problems are evaluated. From 
a mathematical point of view these complete sets of orthogonal functions provide 
the basis for symmetric and biorthogonal transform theorems. Several appli- 
cations of the previous results to the representation of the field of a vertical 
electric dipole in stratified regions aie presented in Section 5. 

2a. Special Field Representations 

In this section we shall be concerned with the representation problem 
posed by the vector field equations (1.1) and its reduction to a scalar problem 
of the general Sturm-Liouville type. Let us first consider the special case 
wherein the excitation is characterized by (p independent, radially directed, 
electric and magnetic current densities, Jr and M^ . Under these circumstances 
the vector field equations, when expressed in polar coordinates r, 6, (p, are inde- 
pendent of <p. Hence, they can be separated into the two independent groups: 

E type 



r sm 6 dd 



(2.1a) -^j^rH,=jkeEe 



H type 
1 d 



rsm $ dd 



sin SE^ = -jkfxHr - Mr 



(2.1b) ll,E,=jk^He 

- — rHg -—Hr= jkeE^ . 

r dr r dd 

Equations (2.1) can be put into a form wherein transmission along either the r 
or the 6 directions is emphasized. 



266 (S266) ELECTROMAGNETIC WAVES 

For the case of r-transmission the transverse (to r) parts of equations (2.1a 
and h) can be rewritten on ehmination of Er and Hr respectively, as 

E type 



(2.2a) 



1 a ^ .J . 1 d 1 d . \„ , 1 dJr 



H type 
1 d 



r dr ^^^ " ^^^^' 



(2.2b) 



11 
r dr 



rHe = jk\ e + jtt- t^ - — z "^ sin ^ £;^ " T" ^ ^^r 
•^ L kr n ddsm e dd J ^ jkn dS 



The longitudinal components E'^ and i7, then follow from the transverse com- 
ponents by the first of equations (2.1a and b), respectively. Differentiability of 
Jr and Mr with respect to 6 is assumed in equations (2.2). 

Alternatively, for the case of ^-transmission it is convenient to express the 
transverse (to 6) parts of equations (2.1) in the form 



1 ^ r. 



E type 

1 



k r dr e dr J ^ 



(2.3a) 



^ ^ sin dH, = jkeEr + Jr 



r sm e dd 

H type 

^^sin BE, = jkfxHr + Mr 

rsm 6 dS 



(2.3b) 



I d ^ .T , 1 d 1 a 1^ 



with the longitudinal components Ee and He determined from the second of 

equations (2.1a and b), respectively. 

2a 1 . r-Transmission Formulation 

Explicit solutions of equations (2.2) in a spherical region ?% < r < r^ , 
02 < 6 < 01 can be obtained in the form of a representation involving ortho- 



NATHAN MARCUVITZ (S267) 267 

normal functions of either B or r. In the former case, now to be discussed, one 
employs the transverse field representations 

E type 

rEeir, ^) = Z V^irMO) 

(2.4a) 

rH,{r, 6)= Z I^m 

H type 

rE,(r, 0) = Z V7(rW{e) 

(2.4b) 

rHe(r, 0) = Z IV{T)k\e) 

i 

where here and in the following the summation sign signifies either summation 
over a discrete index i or integration over a continuous index 2, or both. 

The orthonormal functions e' , hi are defined in such a manner as to simplify 
the determination of amphtudes V[ , 7 • from the E type transverse equations 
(2.2a). The desired simplification is obtained by defining /i' = 6 • , with the h[ 
functions determined by the scalar eigenvalue problem 

(2.5a) fe -^ :^ sin ^ + liAh'^ = 0, 

\ad sm do I 

the eigenvalues /cf being determined by subjecting the h'i to the boundary 
conditions 

(2.5b) \U, ^ (sin 0^9 - W, ^ (sin m,) ^'' = 0,' 

which imply that both U^ and h'i satisfy the same boundary conditions. 

Similarly, the orthonormal functions e-', ^^ are so defined as to simplify 
the evaluation of the amplitudes F-', I'i from the transverse U type equations 
(2.2b). In this case the desired simplification occurs by defining e'i = h'/, the 
e^ being determined by the eigenvalue problem 

(2.6a) (:! --^ ^ sin + I'-^)^- = 0, 

\dd sm 6 do I 

with the boundary condition 

(2.6b) \e, ~ (sin de7) - eV ^ (sin BE,)^'^ = 0. 

The latter imply that both E^ and e[' satisfy the same boundary conditions. 

^[ l^i represents the difference between the values of the bracketed quantity at d-^ and 
at di . 



268 (S268) ELECTROMAGNETIC WAVES 

Equations (2.5) and (2.6) constitute Stlirm-Liouville problems whose explicit 
solution requires the specification of the boundary conditions on the fields E^ 
and H^ ; these conditions must be such as not to destroy the separabilit}^ into 
the E and H type equations. For example, if the spherical region under con- 
sideration is bounded by two perfectly conducting metallic cones of apertures 
6-2 and (9i(f^ 0, tt) 

(2.7a) E^ = = ^ (sin BH^) Sit 6 = 6, , d^ 

ou 

provided Jr vanishes on the conical boundaries. For "proper" eigenf unctions 
(cf. Sec. 3) the boundary conditions (2.5b) and (2.6b) then reduce, respectiveh', 
to 

(2.7b) Sit 6 = di , $2 . 

e'/ = 

By a conventional argument it then follows from equations (2.5-6) that 
both the eigenf unctions e'i and e'/ are orthogonal and can be so normalized as 
to possess the properties (cf. Sec. 3) 

(2.8) / e,((9)ey(^) sin 6 dd = 5,y .' 

As a further example consider the case ^i = 0, ^2 — tt corresponding to 
an unbounded spherical region. In this singular case the boundar}^ conditions 
on the fields are, in view of the finiteness of the latter, 

•\ •\ 

(2.9a) — (sin OE^) = = — (sin dH^) &t 6 = 0, tt. 

ou ou 

The associated boundary conditions (2.5b) and (2.6b) on the "proper" eigen- 
functions are then 

£ (sin dhd = 
(2.9b) at ^ = 0, TT 

^ (sin ee\') = 

from which by means of equations (2.5-6) the orthogonality properties (2.8) 
with 6x = 0, 62 = TT likewise follow. 

The format and notation employed in the representations (2.4) and the 



3 1 i = j 

8ii = if 



i 9^j 



NATHAN MARCUVITZ (S269) 269 

eigenvalue problems (2,5-6) although applied to single-component, i.e. scalar, 
representations are capable of generalization to the two-component vector repre- 
sentations necessary when d/d (p ^ 0. Since the scalar nature of the E- and 
-H'-type representation can still be retained in the latter case (cf. Sec. 2b), it 
is desirable to rephrase the eigenvalue problems of (2.5-6) in accord with the 
format for the general case. One introduces scalar mode functions 0, and xpi 
by the detailed first order equations 

E type ie' = W) \ 



A 
de 

(2.10a) 



= -k'S- 



— (sm ee[) = K •0,- 



sin 6 

H type {e'' = N') 

(2.10b) 

-^^-^(sin BhY) = ^KY^i . 

On elimination of 0^ and \l/i it is evident that equations (2.10a and b) are com- 
pletely equivalent to equations (2.5a) and (2.6a). Alternatively, on elimination 
of e ■ and K defining equations for 0, and ^pi can be obtained in the form of the 
second order equations 

which admit unique solutions when ^i and ypi are subject to suitable boundary 
conditions. For the case contemplated in (2.7b) these conditions reduce for the 
' 'proper" eigenf unctions to 

4>i = 0,' ^ = ^t ^ = ^1.2 . 

whereas the conditions (2.9b) require that </>, and i/^, be finite at the singular 
points ^ = 0, TT. From (2.8) or directly from (2.10c, d) it then follows that the 
</)i and ypi possess the orthonormality properties 



^This condition must be modified when /?,' = 0. This latter possibiUty arises in coaxial 
structures and characterizes the so-called principal mode. For this mode the condition 0,- = 
is replaced by a condition of constancy of 0t/'?t on the various peripheries of the cross-section. 



270 (S270) ELECTROMAGNETIC WAVES 

(2.11) / ct>,{d)(t>j{d) sin ede = 8ij . 

The solution of the scalar eigenvalue problems (2.10c and d) together ^dth the 
relations (2.10a and b) usually provide the simplest procedure for determining 
the mode functions e- and e[\ 

In view of the orthogonality properties (2.8) of the mode functions the 
Vi , li amplitudes in (2.4) can be readily expressed in terms of the fields by 
relations of the form (omitting the distinguishing superscripts) 



(2.12) 



/%0 2 

Vi{r) = / rE{r, e)e,{e) sin d dd 
I.{r) = rH(r, e)h,{e) sin 6 dd. 



The defining equations for Vi{r) and /,(r) can be obtained by transformation of 
equations (2.2) in accordance with the operations indicated in (2.12). Thus, 
multiplying the £^-type equations (2.2a) by e[ sin B and the H-iy^Q equations 
(2.2b) by e-' sin B and integrating over*0 from B^ to B2 , one obtains on use of 
equations (2.5,6, and 12) 






(2.13a) 







";;-i..F.F. + ^, 


where 




K. = {k^e^ - 7jrr^; 


for the E-modes 






Z'i = 


1 


ke 



(2.13b) v: = v:(r) = -Z: r rJXr, ^)e.,((9) sin B dB 

i'i = 
whereas for the ^-modes 

Z7 = (l/YV) = {ky^/K^') 

(2.13c) v[' = 

iY = i[\r) = -Y'i' \ rMr{r, B)Ki{B) sin B dB 



NATHAN MARCUVITZ (S271) 271 

The superscripts distinguishing the mode type have been omitted in equations 
(2.13a) since the equations have the same form for all modes. The functions 
eri{6) and h^iid) are defined by the relations 



(2.13d) 



JK'Ai = K'^i 


rsm 6 do 


JK'Ai = K^^i 


r sm e de 



their significance is evident when it is noted by equations (2.1) and (2.4) that 
the r components of the field may be represented as 

i 

in those regions wherein J^ = = Mr . 

Equations (2.13a) being of conventional transmission line form, constitute 
the basis for terming the amplitudes Vi and li mode voltages and currents, 
respectively; the inhomogeneous terms Vi and ii , characteristic of the excitation 
of the i-ih. mode, are correspondingly designated as the source voltage and 
source current, respectively. The indicated variability with r of the propagation 
wave number k, and of the characteristic impedance Z, implies that equations 
(2.13a) are spherical transmission line equations characteristic of the r variation 
of spherical waves. The wave character is made more explicit by casting (2.13a) 
in the form of second order equations. Thus, eliminating Vi from (2.13a), one 
obtains for the case of the £'-mode wave equation 

whereas eliminating ![' from (2.13a), one finds for the /Z'-mode wave equation 

The amplitudes ¥[ and I'/ readily follow by (2.13a) from the solutions 7' and 
V'i of (2.14). Explicit solutions of the inhomogeneous equations (2.14) will be 
discussed in Sections 3-4. 

With the knowledge of the eigenfunctions e,- and eigenvalues k? from (2.5-6) 
the problem of finding the solution of the partial differential equations (2.2) in 
the form of the representation (2.4) is reduced to that of solving the set of 
ordinary modal equations given in (2.13) or (2.14). Since the solution for one 
mode is typical for every other, the latter problem is solved as a function of 
the mode index i. Although the representation so obtained constitutes a formal 



272 (S272) ELECTROMAGNETIC WAVES 

solution, its practical usefulness is dependent on the rapidity of convergence 

of (2.4). 

2a2 . 6-Tmnsmission Formulation 

Explicit solutions of the <p independent field equations (2.2) in a spherical 
region r2 < r < r^ , $2 < 6 < Bi can equally well be obtained by emplo\^ng a 
representation involving orthonormal functions of r. However, the resulting 
^-transmission formalism does not possess the same general character as that 
for r-transmission because of the lack of vector separability of the field equations 
in directions transverse to 6. For the ^-transmission development one starts 
with the (transverse to 6) field representations 

E type 
r'Erir, ^) = Z Vme'^{r) 



(2.15a) 



(2.15b) 



H type 
r'HXr, d) = J2 IV{e)hY{r) 



rE,(r,d)= Z^^eVir). 



The orthonormal functions e- , M in (2.15a) are so defined as to simplify 
the evaluation of the amplitudes Vi , I'i from the transverse equations (2.3a). 
The desired simplification is achieved by defining h[ = — ee ■ with the functions 
h'i determined by 

(2.16a) (lll+^^^-iy-O 

and the boundary conditions^ 



(2.16b) 



\[^lrH^-rH^ln') 



= 0. 



Equations (2.16) constitute an eigenvalue problem whose solution yields a set 
of orthogonal eigenfunctions hi and associated eigenvalues ac-". Conditions 
(2.16b) apply when e is a continuous function of r. For discontinuous e supple- 
mental conditions of continuity of (1/e) (d/dr) hi and hi are also necessary, the 
latter being a consequence of the continuit}^ of (1/e) {d/dr)rH^ in spherically 
stratified regions. Equations (2.16b) imply that the /? • obey the same boundary 
conditions as i^^ . 

°[ ]!^ ^represents the difference between the va ues of the bracketed quantity at ro and 

at r, . 



NATHAN MARCUVITZ (S273) 273 

The orthonormal functions e-', K' in (2.15b) are defined so as to permit a 
simple evaluation of the amplitudes F-', ?»'' from equations (2.3b). In this 
case simplification results if one defines yu/i-' = e[' , with the mode functions e'/ 
defined by the equations 

and the boundary conditions 

The solution of this eigenvalue problem yields a set of orthogonal eigenfunctions 
e'i' and eigenvalues k'/"^. In analogy with (2.16b) the boundary conditions 
(2.17b) on the e'i are applicable only when /x is a continuous function of r. For 
discontinuous ix there are additional conditions of continuity on (1/m) {d/dr) e'i 
and e •' at the discontinuity points. 

The general Sturm-Liouville problems defined in (2.16) and (2.17) lead to 
unique sets of orthonormal eigenfunctions only on specification of the boundary 
conditions on the fields E^ and H^ . For example, if the spherical region under 
consideration is characterized by continuous e, ii and is bounded by perfectly 
conducting spherical segments at r^ and rs , 

(2.18a) J5:^ = = {d/dr)rH^ at r = n , r^ 

Hence, conditions (2.16b) and (2.17b) reduce for ^'proper" eigenfunctions (cf. 
Sec. 3) to, respectively, 

(2.18b) {d/dr)M = and e'/ = at r = nr^ . 

Alternatively, the spherical region may be open at both ends — i.e. r^ = 
and r2 = CO , The latter are singular points of the differential equations (2.16a) 
and (2.17a) and hence the dependence of the solutions on the boundary condi- 
tions is somewhat different than at regular points. Weyl [2] has distinguished 
between two kinds of singular points: one yielding solutions of the ''limit point" 
type, the other of "limit circle" type. Loosely stated, these terms imply an 
independence of the solutions on the boundary conditions in the former case, 
and a regular dependence in the latter. Since r = corresponds to the limit 
point case, the boundary condition at r = may be stated simply as a condition 
of finiteness of the mode functions and their r derivatives. On the other hand 
r = CO corresponds to the limit circle case and hence the boundary conditions 
at r = CO must be more definite. Since the fields of physical interest usually 
satisfy the ''radiation conditions" 



(2.19a) |^|. + yfc(^,)./^ J^' _ 



as r 



274 (S274) ELECTROMAGNETIC WAVES 

the corresponding conditions on the '^proper" mode functions are by f2.16b) 
and (2.7b) 

(2.19b) |; -f jA:(Me)^'M'' -^0 asr->co 

assuming that fx, e are constant in the hmit r -^ oo . 

With the specification of the boundary conditions on the fields the Sturm- 
Liouville problems in (2.16) and (2.17) become completely defined. Although 
the explicit solution of such problems will be deferred until Section 3, it is de- 
sirable to point out at this point the orthogonality properties of the mode func- 
tions. Thus for the case contemplated by the boundary conditions f2.18), it 
follows from (2.16) and (2.17) by a conventional argument that 

(2.20a) - f e[{r)WAT) ^ = 5„- 

and 

(2.20b) r e7(r)hy(r) % = 5,,- 

plus the other forms obtained from the identities h' = —ee' and ^h" = e"' 
In view of the orthonormalit}^ properties (2.20) the F, , 7, amplitudes in the 
representation (2.15) can be expressed in terms of the fields as 

(2.21a) Vm = - [' E^(r, e)hKr) dr; ^ = - /" ^.C'', Sy,{r) ^ 

and 

(2.21b) me) = f" HXr, e)eV{r) dr; ^^ = [" EM e)h','{r) ^ ■ 

The defining equations for the mode amplitudes T', , /, can now be de- 
termined from the transverse fields equations (2.3) by utilizing the transforma- 
tion relations (2.21). Thus, multiplying the ^-type equations (2.3a) b}' e e-r or 
e ■ and the H type equations (2.3b) by hY or n hY r and integrating over r from 
Vj to r2 , one obtains 



(2.22a) 



where for E'-type modes 



^ = ^,z,/,. + vm 



j^=jlY,V, + Ud) 



z\ 



NATHAN MARCUVITZ (S275) 275 

1 k' 



Y\ k sin d 

(2.22b) z5:((9) = 



r.(e) = sin [ Jr(r, e)e[{r) dr 



and for H-type modes 



^,, _ 1 k sin ^ 



(2.22c) 



vVie) = sin (9 [ M,(r, 0)/irW dr 



t'i\e) = 0. 

Equations (2.22a) have the form of transmission Hne equations, one for 
each mode. The indicated variabihty with B of the characteristic impedances 
Zi imphes that (2.22a) describe wave propagation on ''angular" transmission 
hnes whose propagation wave numbers Ki are determined by the eigenvalue 
problems in (2.16-7). As in equations (2.13) Vi , li are designated as mode 
voltages and currents, while Vi and ii are termed source voltages and currents. 
It is frequently convenient to cast the first order transmission line equations in 
the form of second order wave equations. Thus, by elimination of 7- from 
(2.22a), one has for the £'-type modes 

(2.23a) (-:h#.sin d j-, + l^f)^ = -j^^Z%{d), 



Ksin e de de 
and for the //-type modes by eliminating V'i 

(2 .23b) (-T^ :^ sin -^ + ;r')/r 

\sm 6 de de I 



-jKVYYvm 



The knowledge of the mode functions e^ and of the eigenvalues k^ reduces 
the problem of finding solutions of the partial differentials equations (2.3) to 
that of solving the ordinary differential equation problems posed in (2.22) or 
(2.23). The analysis for one mode is typical of that for all the others; the desired 
solution in the form (2.15) is then found by synthesis of the modal solutions. 

2a3 . Potential Formidations 

A procedure, alternative but intimately related to those discussed above, 
for obtaining a solution of the (p independent field equations (2.1) employs a 
representation not of the fields themselves but rather of potential functions 
from which the fields are derivable. In this so-called method of Hertz (or 



276 (S276) 



ELECTROMAGNETIC WAVES 



Debye) potentials [4] the scalar nature of the representation is introduced at 
an early stage. The method will be treated quite briefly at this point since it 
is taken up in more detail in Section 25. For the £'-type case a potential func- 
tion n'(r, $) is introduced by expressing H^ by 



(2.24) 



tH^ = 



ee 



w. 



By equations (2.1a) one then finds 



(2.25a) 



1 «9^ 
jke dd dr 



^'=i{''^+i:hy 



with n' given by 

(d_ld 
\dr e 



(2.25b) 



1 1 



Ir+^'^^^P^iteie'^'fe 



)„.=-i 



and subject to boundary conditions that follow by (2.24) and (2.25a) from corre- 
sponding conditions on the fields. 

Representations of n'(r, 6) can be obtained in terms of orthonormal func- 
tions of either r or 6, In the former case one employs the representation 



(2.26) 



U'ir, d) = J2 I'ir) 



Ue) 



where the 4>i(6) and k- are the eigenfunctions and eigenvalues defined in equa- 
tions (2.10c). In view of the orthonormality properties (2.11) of the 0^ one has 



(2.27) 



/:-(r) 



Jl'{r, e)(i>i{e) sin e dd. 



To determine explicitly the amplitudes 7' one multiplies equation (2.25b) by 
Ki<f)i sin 6 and integrates over 6 from 6i to 62 . One then obtains b^^ (2.10c), 
(2.27), and the boundary conditions on 0, expressed in the form 

(2.28) [,i„,(,..M:_n'A^..)J- = o, 

the modal equations 

(2.29) (t- - T + ^V - -^i: = - ^ r Jr<i>i sin 6 dd 

\dr e dr er / e Je^ 

for the amplitudes /• . Equation (2.29) is seen to be identical with equation 
(2.14a) previously determined. 



NATHAN MARCUVITZ (S277) 277 

Alternatively n'(r, B) can be represented in terms of functions orthonormal 
in the interval r-^ < r < ?^2 by 

(2.30) n'(r, B) = Y. He) ^'^""^ 



K?/jk 



The mode functions h'i are the same as those defined in (2.16a). From the 
orthonormality properties (2.20a) of the functions h[ = — e e^ the mode ampli- 
tudes Vi follow as 



(2.31) vm = j^ |_ mr, e) ^ 



K{r) dr 



'. V. /oj^ Tirf orrfQ f ir»r» ^rTi-r^n h' 



In accordance with (2.31), multiplication of (2.25b) by h^K^i /jk, integration from 
Ti to r2 , use of (2.16a) and the boundary conditions on /i' in the form 

[K«f-"-f)]>» 

yields as the defining equations for V'i 

(2-33) [^ei^^ 'i + ^^^' = I £" «^'^' *• 

Equation (2.33) is identical with the previously obtained equation (2.23a). 

Since the ^-type equations (2.1b) follow from the ^-type equations (2.1a) 
on the duality replacements Er^>^Hr,E0—>He,H^-^E^ , Jr -^ Mr , e ^ ju, 
and ju -^ e, it is unnecessary to repeat the details of the above formalism for 
the ^-type case. One has only to introduce into equations (2.24-33) the duality 
replacements W -> n'', /; -^ V'/, Y'i -^ I[', 0, -^ i^^ , kJ -^ k<', Wi -^ -e\', and 
e- — ^ In'i to obtain the /Z^-type formalism. 

2b. General Field Representation 

The solution of the field equations (1.1) in an r-st ratified spherical region 
with arbitrary excitation J and M is facilitated by elimination of either the r 
or B field components. Only the elimination of the r-components, corresponding 
to an r-transmission analysis, will be considered here. In this case the resulting 
transverse equations may be cast in an invariant (two) vector form that can 
be obtained by vector and scalar product multiplication of equations (1.1) by 
the radial vector r. These operations yield, respectively, 

V(r-E) - (r- V)E - E = -jkr X H - r X M 
(2.34a) 

V(r-H) - (r- V)H -H= jkr X E + i X J 



278 (S278) ELECTROMAGNETIC WAVES 

and 

V-r X E = jk^iT'll + r-M 
(2.34b) 

V-H X r = jkeT'E + r- J 

on use of simple vector relations and the fact that V X r = 0. On elimination 
of the radial field components from (2.34a) by means of (2.34b), the transverse 
equations follow, as 

- |;rE, = jk(^„ + p^ .VV.)-H X r + M X r + ~^ 



(2.35) 



-^H'=4 + P;;'VV.)-rXE,+rXj + ^^ 



where the subscript t denotes vector components transverse to the r direction. 
In accordance with this notation the vector gradient operator V has been 
decomposed either as 

(2.36a) V = V. + ^ |:r% = V - (V -10)10 

or as 

(2.36b) v = ,V+ro|;=V- ro(ro- V). 

The necessity of distinguishing between the transverse component V< taken 
from the right and «V the transverse component from the left is a consequence 
of the variability in direction of the unit radial vector To . 

To evaluate in scalar terms the transverse fields defined b}^ the vector 
partial differential equations (2.35) one can proceed in either of two ways. The 
vector fields themselves can be represented as a superposition of an infinite 
number of characteristic vector modes, each vector mode being then decomposed 
into two components E and H modes whose scalar amplitudes are determined 
by ordinary differential equations. Alternatively, the transverse vector fields 
can be represented in terms of two scalar potentials that in turn can each be 
represented as a superposition of an infinite number of characteristic scalar 
modes whose amplitudes are also determined by the same ordinary differential 
equations. The former procedure was adopted in subsection 1 of Section 2a, 
the latter in subsection 3 of Section 2a; for the special (p independent excitation 
Jr and Mr treated therein it was possible to effect a natural separation into the 
component E and ^-modes at all stages of the development. The lat4:er pro- 
cedure will be employed throughout most of this section in a rather general 
form; it is essentially the method of Hertz or Debeye potentials and is intimately 
related to the Green's function techniques discussed in Section 3, et seq. 

Any transverse vector can be decomposed into a transverse gradient and 



NATHAN MARCUVITZ (S79) 279 

a transverse curl part. Thus, to obtain the desired scalar reformulation of 
equations (2.35), let 

E, = -,VF' - ,V X V'% , L = -tVJ' - .V X J'% 

(2.37) 

H, = -.Vr' + .V X Pto , M, = -,VM'' + .V X M'ro 

where the scalar functions (of r, 6, <p) V , V" are constitutive measures of E< ; ; 

r, /" of Hj ; etc. The scalar functions /' and V" appear as generalizations of 

the potentials 11' and Jl", respectively, of Section 2a. On substituting (2.37) 

into (2.35), noting that the operators r<V and d/dr commute, and equating the i 

independent transverse gradient and curl terms, one obtains as the defining '; 

equations for the potentials, the set of scalar equations ! 

E type jlj 



V = ,.(m + ^)/' + M' - ^ 



(2.38a) 



-|;/' =ifeF' + J' 



and H type 



(2.38b) 



dr 



'<-*wy^'"-s. 



where V r < V is the two dimensional operator that in polar coordinates 6, is 
given by (2.36) as 

(2.39) ^'Vr.V = ^^sin0^ + ^4-^ . 

sm ^ a^ dd sm d d(p 

The solutions of the simultaneous partial differential equations (2.38a) and 
(2.38b) are to be subjected to as yet unspecified boundary conditions on V\ I' 
and V", I". For example for a perfectly conducting metallic boundary limiting 
the cross-sectional region By < B < 62 and ipi < v? < ^2 , it follows by (1.2) 
and (2.37) that the boundary conditions are 

v = = r 

(2.40) 

^11 = = 11 

dv dv 

where v is normal to the boundary and perpendicular to the r direction. For 
an unbounded B, (f cross-section the boundary conditions on V, I' and Y" , 



280 (S280) ELECTROMAGNETIC WAVES 

I" are simply finiteness and single- valuedness. The cross-sectional boundary 
conditions are subject to restrictions that make valid the E and H decomposition 
implied in equations (2.38a and b); this decomposition is valid for the boundary 
conditions just mentioned. The remaining longitudinal boundary conditions are 
determined by ''impedance" conditions in the r direction. For example, at a 
perfectly conducting surface with specified r 

V = = V" 

dr dr ' 

for a radiation condition as r -^ oo 



I' - \e) - I' 



which implies that there are no sources and e, fi are constant as r — » oo . 

The partial differential equations (2.38a) can be reduced to a set of ordinary 
differential equations by employing a representation in orthonormal functions 
characteristic of the operator r^ V « • < V . Thus let 

V'ir, 9, ri = Z VKr) ^^ 

i Ki 

(2.41) 

i Ki 

where, in view of (2.38a) the characteristic mode functions <^, are defined by 
the eigenvalue problem 

(2.42) (r'Vr.V+ kO0, = 

with the eigenvalues Icf determined by subjecting (pi to appropriate boundary 
conditions. The mode functions 0,- defined in (2.42) possess by the usual argu- 
ment the orthogonality properties 

(2.43) // 0A- dQ = 5„- 

where the surface integral with respect to d^ = sin 6 dd d<p is extended over the 
spherical cross section (perpendicular to r) of the given region. From (2.41) 
and (2.43) it follows that the mode amplitudes are given in terms of V'{r, d, tp) 
and 7'(r, 6, ^) by 

Vr{r) = ^K'i jj V'(j>i dn 

(2.44) 

Fiir) = K< jj r<t>i dQ. 



i 



NATHAN MARCUVITZ (S281) 281 

To obtain the defining equations for the mode amplitudes by transforma- 
tion of equations (2.38a) in accordance with (2.44), it is necessary to employ 
Green's theorem for 0,(^, (p) and /'(r, 6, (p) in the two dimensional form ap- 
propriate for a spherical surface, namely: 

(2.45) Jl [0,Vr.Vr - r\/rtV<l>iV dU = j [0, ^ " ^' ^J ds' 

The left hand surface integral is extended over the spherical cross section of 
the given region, whereas the right hand line integral with respect to ds is taken 
over the peripheral curve s, if any, bounding the cross-section; as before v 
denotes the outward normal direction at s. Equation (2.45) assumes a particu- 
larly simple form if the boundary conditions on the <j>i are so chosen that 



(2-^'^) ^f-^'f] = o 



on s. 



Multiplication of equations (2.38a) by 0^ and integration over the cross-section 
then yields by (2-44-6) as the defining equations for the £^-mode amplitudes 



(2.47) 



dV 



- 'f = j4Y',v: + ii 



where 

v',(r) = ;; fj M'(r, e, ,p)U6, v) da - ^^ ff J.{r, 8, v>)US, v) dO. 

i'i{r) = 'i^i jj J'ir, e, ,p)<t,tie, ^) da 

The partial differential equations (2.38b) can be reduced to a set of ordinary 
differential equations in a manner similar to the above. In this case one employs 
the representations 



6d/dv= VfV 



282 (S282) ELECTROMAGNETIC WAVES 

V"{r, e,^)=T, V'Ar) ^^%^ 

i Ki 

(2.48) 

where the characteristic modes ^»- and the characteristic values k'/ are defined 
by the eigenvalue problem 

(2.49) (rVr.V + ^K'/')rP, = 

with \f/i subject to suitable boundary conditions. The orthonormality properties 
of xpi on the spherical cross-section, 

(2.50) jj lAi^A,- dU = 8a , 

are readily deduced with the consequence that in (2.48) 

V7{r) = ;:•' // V'\r, d, <p)xl.^(d, <p) d^ 
(2.51) 

mr) = ^r // r'{r, e, ^)^.{e, <p) dn. 

The transformation of equations (2.38b) in accord with the operations indicated 
in (2.51) then yields on use of the implied boundary conditions on \l/i , 

(2.52) ^^-^"17]=° °^'' 

the defining equations for the i^-mode amplitudes: 

_dVl 
dr 
(2.53) 

dr/ 

dr 
where 



= JK'/Y'/VV + i'/ 



'Vir) = «:' // M"{r, e, v)Ue, <P) da 

iV{r) = 'kV ff J"{r, e, <p)ue, v)da-^ ff j^r, e, v)Ue, <p) da 



ff" /^2\l/2 



(F€M - k/ /r') 



rv,f L _ ^ 



J 



NATHAN MARCUVITZ (S283) 283 

Equations (2.47) and (2.53) are spherical transmission line equations of a more 
general form than the similar equations (2.13a) encountered in the <p independent 
analysis of Section 2ai . The chief difference between the two cases lies in the 
greater complexity of the source voltage Vi and source current ii in the former 
case of arbitrary excitation. 

Although the transmission line equations have been obtained in this section 
by a scalar representation of the potentials /', Y" ^ etc., it is perhaps of interest 
to sketch the derivation of equations (2.47) and (2.53) by direct vector repre- 
sentation of the fields. In this case one starts with the transverse vector repre- 
sentations 



(2.54) 



rH,(r, fl, «,) = Z Vl'lr)K{S, <p) + I'/irW/ie, <p)] 



where the single and double prime superscripts denote, respectively, the separa- 
tion into E and H vector mode functions; this separation implies a restriction 
on the boundary conditions permissible on the cross-section of the spherical 
region under consideration. The vector mode functions in (2.54) are to be so 
chosen as to permit a simple evaluation of the mode amplitudes. Although the 
desired functions are evidently characteristic of the operator « V V < in equations 
(2.35), we shall omit for brevity their detailed derivation and merely define 
them in terms of the already defined scalar mode functions 0i(^, (p) and \l/i{d, (p). 
For the spherical regions in question the £'-mode functions are defined by 

(2.55a) , hj = To X e^ 

and the ^-mode functions by 

-;-hr = r,v^, 

(2.55b) , er = hr X To . 

It is evident that the detailed equations (2.55) are equivalent to the scalar 
eigenvalue equations (2.42) and (2.49) and are generalizations of the corre- 
sponding (p independent equations (2.10a) and (2,10b). Moreover, from (2.55) 
one obtains on elimination of 0, and xpi , respectively 

(2.56a) r\VVt-e'i + I-f e^ = 

(2.56b) rWVrhy + Ic^'h^ = 

which, together with appropriate boundary conditions, constitute the vector 
eigenvalue problems for the E and ^-modes, respectively. From equations 



284 (S284) ELECTROMAGNETIC WAVES 

(2.56), or from (2.55) and the previously mentioned orthonormality properties 
of 4>i and xpi , one readily deduces 

jj e^e; dn = d,j = jj er-er dn 

(2.57) 

dn = 



Ih'-^'' 



together with corresponding orthonormality properties for the h, ; as above the 
surface integrals are to be extended over the entire cross-section. From (2.54) 
and (2.57) one then finds that 

F, = jj rEre.da 
(2.58) 

I, = jj rU, h, d^ 

for both mode types. Transformation of equations (2.35) according to (2.58) 
then yields, after vector integration by parts, the same spherical transmission 
line equations (2.47) and (2.53) for the E and H mode amplitudes of (2.54). In 
terms of the vector mode functions e, , h^ the source terms are given by 

Viir) = jj rM-hid^ - Zi jj rj-e^idn 
(2.59) 

i.{r) = jj rj-e, dQ - F, jj rM-h,, d2 

where the r component vector functions e^i and h,, are defined by 

Equations (2.59) could equally well have been obtained from the corresponding 
expressions in (2.47) and (2.53) on use of (2.55); this involves an integration 
by parts assuming M' = = J' on the cross-sectional boundary. The modal 
superscripts have been omitted in equations (2.59) since the equations have the 
same form for both mode types providing one notes that for jEJ-modes h,, = 
whereas for ^-modes e^^ = 0. It should also be noted that the expressions for 
Vi and ii in (2.59) do not require the explicit decomposition of J and M into 
their E and ^-components. 

3a. Characteristic Problems in One Dimension 

As discussed in Section 2, the solution of a vector field problem, or of 
equivalent scalar problems, in the form of a representation requires the solution 
of both eigenvalue and transmission line problems. Although eigenvalue prob- 



NATHAN MARCUVITZ (S285) 285 

lems are in general multidimensional, their solution frequently may be traced 
back to corresponding one dimensional problems of the type considered in this 
section. From an operational point of view the one dimensional problems of 
interest are characteristic, in general, of a non-Hermitean Sturm-Liouville 
operator 

(3.1) L=-£p(x)£+,(x), 

where 7? and q are assumed to be piecewise continuous in the interval Xi < x < X2 . 
Associated with the operator L is a characteristic Green's function defined in 
the indicated interval by the inhomogeneous differential equation 

(3.2a) [L - \w{x)]G{x, x') = 8(x - x') 

and subject to the boundary conditions 



(3.2b) 



p -^ — h q;i,2 \G(x, x') — > as a; 



The arbitrary complex parameter X is to be so restricted as to insure the unique- 
ness of G{x, x'); the weight function w{x) is a piecewise continuous function; 
and the delta function source term is defined by 

b{x — x') = Q iix 9^ x', I b{x — x') dx = 1, 

the interval of integration including the singular point x' . 

Also associated with the operator L are the set of characteristic functions 
<i)i{x) defined in the interval x^ < x < Xa by the homogeneous differential equa- 
tions 

(3.3a) [L - \iw{^)]4>i{x) = 0. 

The characteristic values X^ are determined by subjecting </>,(x) to boundary 
conditions defined in terms of the corresponding conditions (3.2b) on G{x, x') by 

where the left hand side denotes the limiting difference of the bracketed quantity 
at X ^ 0^2 and x ^>- Xi . The boundary conditions on </>, are phrased in the form 
(3.2b) to include the complete set of both proper (discrete) and improper (con- 
tinuous) eigenf unctions; the former are square integrable in the given interval, 
the latter are not. For the proper eigenfunctions the boundary conditions 
(3.3b) may be reduced to those of the form (3.2b); this reduction is not possible 
for the improper functions. In view of the existence of finite solutions to 
equations (3.3a) there is a manifest ambiguity in the G{x, x') defined by (3.2a) 
when X = X^ . Thus the uniqueness restriction on X alluded to above is that 
\ 9^ \i . For real p, g, w, and ai.2 (the Hermitean case) it can be shown that 
the eigenvalues X^ of the operator L are real; hence for this case the restriction 



286 (S286) ELECTROMAGNETIC WAVES 

X 7^ X, reduces to ^m \ 9^ (provided the X plane can be regarded as a simple 
surface) . 

By comparison with equations (2.14) and (2.23) it is seen that equations 
(3.2) characterize a transmission line problem in which G(x, x') represents either 
the voltage V or the current / produced by a 5 function source at a; = x' . 
Although phrased as a second order differential equation problem, equations (3.2) 
could equally well have been put in the first order form more customary in 
general transmission line theory and more convenient when more general tj'pes 
of sources are treated. It is likewise evident that the characteristic function 
problem of (3.3) is a generalization of corresponding problems encountered in 
equations (2.11) and (2.17). 

As previously stated and as also implied in (3.2) and (3.3) there exists 
an intimate connection between the characteristic Green's function G{x, x') and 
the characteristic functions <f>i{x). As a consequence, the knowledge of G{x, x') 
implies that of the </)^(x), and conversely. This connection has been exploited 
in various ways by Weyl, Titchmarsh (loc. cit.) et al., to obtain the spectral 
representations (and hence the characteristic orthonormal functions) for a large 
number of operators L. In the following we shall employ essentially the same 
reasoning as the above authors to determine explicitly the orthonormal functions 
characteristic of the operators involved in the representation of fields in the 
spherical regions discussed in Section 2. However, the procedure to be employed 
below will involve the 5-function technique, whose oft discussed justification 
will not be considered herein. The virtue of this technique, besides that of 
simplicity, is that questions of ''completeness" are answered naturally, albeit 
formally, by recourse to the basic concept of the 6 function. For example, the 
existence of a complete set of orthonormal functions </),(x) presumes the ability 
to represent completely an arbitrary continuous function in terms of the 4>i{x). 
If this arbitrary function is a 5 function, the completeness and orthonormality 
of the (j>i{x) with respect to the weight function w{x) is contained in the existence 
of the representation 

the summation sign denoting here and in the following either or both a sum over 
a discrete index i characteristic of a discrete spectrum or an integral over a 
continuous index i characteristic of a continuous spectrum. That equation (3.4) 
is a completeness statement for an arbitrary continuous function fi^x) follows 
formally by multiplication of (3.4) by f{x')w{x') and integration over x' from 
Xi to X2 , etc.; orthonormalit}^ follows formally by multiplication of (3.4) by 



^Discontinuous function representations can likewise be considered b}^ a simple "arith- 
metical mean" extension of the following considerations. 

^Other non symmetrical forms of the completeness statement appropriate to Hermitean 
orthogonal and to bi-orthogonal representations will also be employed. In the former case 
<i>i{x*') in (3.4) is to be replaced by <t>i*{x'), in the latter by \pi{x') (an adjoint function). 



NATHAN MARCUVITZ (S287) 287 

(f),(3^')w{x^) and integration from Xi to Xz , etc. It should be noted that the 
representation statement for a function f(x) can be cast in the form of the 
transform theorem 

(3.4a) fix) = T.FMX) 

(3.4b) ^^ = [ ' f(^)^i(^) d^ 

where the sum sign is to be regarded in the general sense stated above. For a 
bi-orthogonal representation 0^ in (3.4b) is to be replaced by t/^, . 

Assuming the existence of the representation (3.4), one can investigate the 
connection between G{x, x') and the <^i{x). For by equation (3.4) the char- 
acteristic Green's function can be represented as 

(3.5a) G{x, x') = E G4i{^) 

(3.5b) with Gi = f ' G{x, x')<t>,{x)w{x) dx. 

J Xx 

The amplitudes G, may be determined from (3.2a), the defining equation for 
G{x, x'), on multiplication by (f)i{x) and integration in accordance with (3.5b). 
On use of the self adjointness relation (Green's theorem) for the operator L 

£ [Ux)LG{x, x') - G(x, x')L<l>,{x)] dx = -\p[.t>< ^-0 g^)J" , 
and of the defining equations (3.3) the (t>i{^), there is then obtained for 

(3.6) G< = - ^^ 

A — Aj 

whence 

(3.7) G(x,:.') = - E¥^- 

i A A,- 

The representation (3.7) obtained on the assumption of the completeness 
of the representation (3.4), indicates that G(x, x') possesses singularities in the 
complex X plane at the points X,- . These singularities take the form of either 
poles or branch cuts depending on whether the X, characterize points of the 
discrete or continuous spectrum, respectively. In a purel}^ formal way one can 
integrate (3.7) about a contour in the X plane enclosing all the singularities of 
G{x, x') and obtain by Cauchy's theorem the basic relation 

(3.8) -^.j G(x, X') d\= Z 0..(,r)<A,(.r') = ^%'^^,f^ • 

^On use of (3.5b) in (3.6) one obtains a homogeneous integral equation for <^, and X, which 
can be employed in a formulation of the eigenvalue problem alternative to that in equations 
(3.3). 



288 (S288) 



ELECTROMAGNETIC WAVES 



I 



The contour is taken around not only the poles of G{Xy x') but also the branch 
cuts if they exist. The rigorous justification of the above procedure has only 
been sketched. For Hermitean operators L a rigorous proof is contained in the 
works of Weyl, Titchmarsh, etc.^° alluded to above. As is implied in the above 
sketch, difficulties in a rigorous proof arise when the operator L admits a con- 
tinuous spectrum. It is then usually necessary to employ a limit process starting 
from the readily handled case wherein the only singularities of G in the X plane 
are simple poles and then pass to the case wherein some or all of the poles coalesce 
into a branch cut or cuts characteristic of the continuous spectrum. 

The contour integral relation (3.8) is the basis of a well defined procedure 
for the solution of the eigenvalue problem associated with an operator L. The 
virtue of this procedure is that the problem of finding a complete orthonormal 
set is reduced to that of constructing the characteristic Green's function G{x, x') 
and completely investigating its singularities. Questions of uniqueness and de- 
pendence on boundary conditions of the characteristic orthonormal set are 
reduced to corresponding, and more easily treated, questions for a single char- 
acteristic Green's function satisfying a well defined inhomogeneous differential 
equation. 



3b. The Characteristic Green s Function (Resultant) 

The explicit construction of characteristic Green's functions involves the 
solution of one dimensional transmission line problems similar to those posed in 
Section 2. Equations (2.13a), for example, exhibit such problems in the general 
form of two simultaneous first order equations subject to boundar}^ conditions. 
For arbitrary source distributions v and i it is frequently convenient to reduce 
the first order equations to second order equations of the form sho^^^l in (2.14a) 
or (2.14b) by means of a superposition argument, assuming first z = and then 
V = ^. The latter equations are characterized b}^ the operator L of (3.1) and 
hence their solution may be reduced, again utilizing the superposition argument, 
to the solution of the general Green's function^^ problem stated in equations (3.2). 

For illustration, if in (3.2) with x = r and 

(3.9a) j){x) = -^ , q{x) = -k'n{r), X = -k? , wix) = ^ , 

the Green's function solution is designated as Gi(r, r'), the solution of (2.14a) is 
given by superposition as 



(3.9b) 



,.(r) = jk f Gj{r, ry^ir^) dr', 



loioc. cit. Also cf. K. O. Friedrichs, "Spectral Representations of Linear Operators", 
lecture notes, New York University, 1948. 

"If X in (3.2) is an arbitrary complex parameter, G{x, x') is designated as a ''characteris- 
tic" Green's function (i.e. inverse operator (L - X)"^) ; if X is fixed the adjective "characteristic" 
is omitted. 



NATHAN MARCUVITZ (S289) 289 

where the irrelevant prime superscript and the hmits of integration have been 
omitted. Correspondingly if in (3.2) with x = r and 

(3.10a) p{x) = -TT , qix) = -k\{r), X = -k- , w{x) = -rp , 

the Green's function is designated as Gv{t, r'), the solution of (2.14b) (omitting 
the double prime superscript, etc.) is given by 

(3.10b) V,{r) = jk j Gv{r, t')u{t') dr\ 

It has been tacitly assumed that the boundary conditions (3.2b) on Gi{r, r') 
correspond to those on 7, whereas the conditions on Gy(r, r') correspond to those 
on Vi ; for open regions wherein both F^ and /» satisfy the same ' 'radiation" 
conditions, Gj = Gv • Since (3.9b) is the solution of (2.13a) with ^ = and 
since (3.10b) is the solution with ?; = 0, it follows by appropriate superposition 
that the general solution of (2.13a) is 



(3.11) 



h{T) = jk f Grir, r')v,{r') dr' - -^ |; / Gy{T, r')U{T') dr' 
F.(r) = jk f Gy{r, r^)Ur^) dr' - — y- | / G.ir, f)v.{r') dr\ 



This explicit solution is manifestly dependent on the solution of a Green's func- 
tion problem of the type shown in (3.2). 

Let us therefore briefly review the method of solution of the inhomogeneous 
differential equation (3.2a) subject to the boundary conditions (3.2b). One 
readily deduces by integration of (3.2a) about an infinitesimal interval centered 
at x', that the presence of the function source is equivalent to the demands that 
p{dG/dx) possess Q,t x = x' sl jump discontinuity of value — 1 and that G be con- 
tinuous Sit X = x'. At all other points x 9^ x' one likewise deduces that G 
satisfies the homogeneous form of (3.2a) and that both p{dG/dx) and G are 
continuous; the latter property is of importance since p, q, and w are permitted 
to have jump discontinuities. A further interesting property of G{x, .t') readily 
derivable from (3.2) is that G{x, x') = G{x' , x) if p, q, w and ai,2 are real or 
complex. This symmetry property facilitates the explicit evaluation of G(x, x'). 
One considers two independent solutions of the homogeneous Sturm-Liouville 
Equation, the first obeying the boundar}'- condition (3.2b) on (r at x = x^ and 
the second that on G at x = Xo . The two solutions are defined by 

(3.12a) [L - \w{x)]T{x) = 0, (?^ ^ + «'V ^ ^ nt x = x, 
(3.12b) [L - \w(x)]U{x) = 0, {p^ + «-'^) = ^ at x = X, . 



290 (S290) ELECTROMAGNETIC WAVES 

In virtue of the above stated symmetry and continuity properties the char- 
acteristic Green's function of (3.2) can be expressed in terms of these solutions as 



T(x)U{x') 
W{U, T) 
(3.13) G{x ,x') = 

T{x')U{x) 
W{U, T) 



X < x' 



X > x' 
where T'F([/, T) the Wronskian expression defined by 

(3.14) ^(t;,r)=p(t/f-rf), 

is independent of x in those regions wherein p, q, w and their deri^'atives are 
continuous. Since (3.13) satisfies (3.2a) and the boundary conditions (3.2b) 
and also possesses the required continuity properties at a; = x', it is apparent 
that it is the desired solution. The discontinuous representation in (3.13) is 
more succinctly stated in the form 

(3.15) Gix, X') = '^\yf^[%''^ 

where the notation x< is employed for either x or x' depending on which is the 
smaller, and conversely for x> . An even more compact notation is obtained if 
in addition T and U are so normalized that the Wronskian (3.14) is unit5\ 

The constancy w^ith x of the Wronskian can be employed to obtain an 
alternative expression for G{x, x') that frequently facilitates the explicit con- 
struction of G{x, x'). Thus, if the Wronskian is evaluated at a convenient 
point, say Xq , and (3.15) is divided by T{xq)U{xq), there is obtained 



(3.16) 


G{x, X') = 


t{x< 


, Xo) M(a;> , a-o) 
-X(X) 


where 










l{X, Xq) — „. s 




u(x X) ^'(^) 

U{X, Xo) — jT( ^ 



^-«--(lf).. ^-wfef).. 

X = x~ + x^ 

General transmission line theory provides a ready phj^sical interpretation 
of (3.16). As evident from equation (3.9), G{x, x') can represent, for example, 
the current wave set up at any point a; by a voltage point som'ce of amplitude 
\/jk at x'. Thus t{x, Xq) and ii{x, Xq) are wave solutions normalized to unit}' 
2X X = Xq , the former satisfying the prescribed terminal (impedance) conditions 



NATHAN MARCUVITZ (S291) 291 

at X = Xi , the latter Sit x = X2 . The quantity X(X) represents the total re- 
actance (= iZ{\) where Z(\) is the total impedance) at Xq and is the sum of 
the reactance X~(\) and X^{\) looking from Xq in the negative and positive x 
directions, respectively. Alternatively, as in equation (3.10), G(x, x') can repre- 
sent the voltage wave set up by a point current source; in this case the de- 
nominator of (3.16) is denoted by B{\), the total susceptance (= -\-iY{\) where 
F(X) is the total admittance) at Xq . The simple zeros of the total reactance 
(or susceptance) define the so called resonances and resonant wave solutions of 
the transmission system; the corresponding X values and distribution functions 
t{x, Xq) are the eigenvalues and eigenfunctions of the discrete spectrum. One 
virtue of introducing impedance terminology in this connection is that the 
presence of a discrete spectrum, i.e. of resonances, have well established and 
intuitive answers in impedance theory. Moreover, the existence of systematic 
methods for expressing the reactance at any point say o^o , in terms of the pre- 
scribed terminal reactances 0:1,2 provides a standardized formalism for con- 
structing the characteristic Green's function of (3.16) almost at once. Although 
admittedly a question of terminology, impedance phraseology clothes a syste- 
matic procedure for obtaining solutions of arbitrary second order differential 
equations. 

Let us return from the above digression to the explicit evaluation of the 
logarithmic derivatives X~ and X^ together with the homogeneous solution 
t(x, Xq) and u{x, Xq) of equation (3.16). These can be expressed in terms of a 
set of regular (standing wave) fundamental solutions 

c{x, Xq) and s{x, Xq) 

satisfying the homogeneous equations in (3.12) and the real boundary conditions 

C{Xo , Xq) = 1 S{Xo , Xq) = (J 

(3.17) 

p{xo)c'{xq , o^o) = p{xo)s'(xo , a;o) = 1 

where the prime denotes a derivative with respect to the first argument. It 
follows from (3.12) that 



(3.18) 



where 



t{x, Xq) = C{X, Xq) — X S(X, Xq) 
U(X, Xq) = C(X, Xq) -j- X^S{X, Xq) 



/g Jgg^N J^- ^ Pi^l) C\X^ , Xq) + a, C(X1 , Xq) 

p(Xi) siXi , Xq) + «! s(.Ti , O^o) 

/q 1 0"U\ Y'^ P(^2) C (^2 , Xq) -\- ^2 C{X2 , Xq) 

P{X2) S{X2 , Xq) + ^2 S{X2 , Xq) 

If Xiixz) are singular points of the differential equation (3.2a), the "limit point" 



292 (S292) ELECTROMAGNETIC WAVES 

or ''limit circle" case is said to obtain at a:i(a;2) when there is respectively an 
independence or regular dependence of X~{X^) on the limiting terminal re- 
actance ai(a2). In terms of the solutions (3.18) the characteristic Green's 
function (3.16) may be written 

/Q in\ nr^ ^'\ — L<^(^< , Xq) — X s(x< , Xq)][c{x> , Xp) + X s{x> , Xp)] 
Vo.iy; ij-yx, x j — —X(X) 

the singularities of which can be inferred from the regularity properties of X" 
and X^ in the X plane. Three cases can be distinguished: 

(1) X~ and X^ Meromorphic 

In this case the only singularities of G are simple poles (since dX'dX 7^ 0) 
located at the zeros X„ of X(X). The completeness relation (3.8) ^-ields on 
evaluation in (3.19) of the residues of Gix, x') at X = X„ 

/Q om ^(^ ~ ^') _ V ^x„(3^, x^ uxn(x\ Xq) v/'\ ^ - n 

(^•^^^ w{x') - ^ {dX/d\),„ ' ^^^"^ - ^- 

The spectrum is evidently discrete. The eigenf unctions normalized to unity 
with respect to the weight function w{x) are 

/q Ono\ ^Xn(^? ^0) ^ h„{X, Xq) 

^^•^''^ Kdx/d\},,r' - Kdx/d\),r' ■ 

(2) Only X^ Meromorphic 

In this case the singularities of G take the form not only of simple poles 
located at the zeros X„ of X(X) but also of branch cuts through the branch points 
of X~(X). The completeness relation (3.8) yields after contour integration of 
(3.19) 

/Q 01 \ ^(^ ~ ^') _ X^ ^Xn(^, Xq) UxSx', Xq) , J_ (£ ^(x< , Xp) l<(x> , .Tp) 

^^•^^^^ w(x') ~ ^ (aXM)x„ + 27r " Z(X) "^^ 

^^•^^^^ ~^ idX/dX),^ +27rr Z(X) ^^ 

where Z(X) = —iX{\). The sum represents the contribution of the residues 
at the poles of G; the contour integral (taken in the clockwise sense) represents 
the contribution from the branch cut. Equation (3.21a) represents the con- 
tinuous eigenf unctions in biorthogonal form; the symmetrical representation of 
(3.21b) is obtained by noting that the X part of the X~ = X — .X^ term in /(.r, Xp) 
does not contribute to the branch cut integral. The normalized discrete and 
continuous eigenfunctions can be readily recognized in the latter representation. 

(3) X~, X^ not Meromorphic 

Both a discrete and a continuous spectrum are possible. The completeness 
statement (3.8) in this case can be written in the same biorthogonal form sho\\Ti 



NATHAN MARCUVITZ (S293) 293 

in (3.21a). However, the symmetrical form^^ of the completeness relation 
becomes 

8{X — X') ^ yy UxSx, Xp) Ux,W , X^ 

w{x') 4' (^x/ax)x„ 

4_ _L (£ / c(^, Xo)c{x\ Xq) s{x, Xq)s{x', Xq) 
'^ 2tJ\ Z{\) "^ F(X) 

2Z00 '^^^' Xo)s{x, Xq) + c(a:;% Xo)s{x, Xp)]} d\ 

where Z(X) = -iX(\) and F(\) = -i{l/X^ + 1/X~). The discrete and 
continuous normalized eigenfunctions follow readily from the representation 
(3.22). 

The above representations are illustrated in more detail in Sections 4 and 5 
for some of the operators L encountered in spherical propagation, problems. 
Representations of the form (3.20-2) apply even when 7?, q and w are discon- 
tinuous functions of x] the consequent discontinuous (in x) nature of the repre- 
sentations can be exhibited explicitly in such cases (cf. Sect. 4). 

4. One Dimensional Spectral Representations 

Several complete sets of eigenfunctions and eigenvalues characteristic of 
the one dimensional operators composing the spherical wave operator V^ will 
be evaluated in this section. The operators in question are distinguished by 
the values of the parameters p, q, w • • • in the operator L defined in equation 
(3.1) and by the boundary conditions characterizing the domain of admissible 
functions. The spectral representations to be obtained will be utilized in 
Section 5. 

4ai . p = sin 6, q = 0, w = sin 6; < 6 < t 

The characteristic Green's function (3.2a) is defined in the indicated domain 

by 

(4.1) [^™ *^ + ^^'"^ *]^(^' ^') =-«(»- »'). 

The boundary points d = 0, it are regular singular points of the differential 
operator and are of the "limit circle" type; however, rather than employ condi- 
tions of the form (3.2b), G(d, 6') is most simply characterized b}'- conditions of 
finiteness at the boundaries = 0, tt. The Hermitean character of L and con- 
sequent reality of the eigenvalues implies that G{d, 6') is unique \i dinX 9^ 0. 
If X = v{v + 1), solutions satisfying the homogeneous equations (3.12) and 



i2cf. Titchmarsh, loc. cit. Eq. (3.1.8). 



294 (S294) 



ELECTROMAGNETIC WAVES 



the required finiteness conditions at ^ = and tt are, respectively, the Legendre 
functions 



(4.2) 



T(e) = PXcos 6) 
U{B) = P,(-cos 6). 



The independence of these solutions for £fm ?/ 5^ is assured by the non- vanishing 
of the Wronskian expression 



(4.3) sin e\ P,(cos 6) j- P,(- cos 6) - P,(- cos B) j- P,(cos 6) 

In accordance with equation (3.15) G{d, B') is thus given by 

P,(cos g<)P,(-cos B>) 



1 2 . 

= - sin vir. 

J TT 



(4.4) 



G{B, B') = - 



(2/7r) sin vw 
whence by (3.8) the completeness relation in biorthogonal form follows as 

(4.5) 



8{B - 6>0 ^ _L(k Pv(cos 6><)PX- cos ^>) - 
sin B' 2'Ki ■^ (2/7r) sin vir 



with the contour integral taken around all the singularities of the characteristic 
Green's function in the X plane. The regularity of the Legendre functions and 
sin VTT in the X plane (note P, = P-v-i) implies that the only singularities are 
simple poles on the positive real X axis at ^ = n = 0, ±1, ±2 • • • , i.e. X = 
{n -\- \Y — \. On evaluation of the integral (4.5) along the contour indicated 



3 



<: 



Figure 4.1 

in Figure 4.1, one obtains by the Cauchy residue theorem as the symmetrical 
form of the completeness statement 

KB - B') ^ y> (n+ ^)P„(cosg<)P.(-cosg>) 
sin^' h (-1)" 

(4.6) 

00 

= Z (^ + i)Pn(cos ^)P„(cos B') 



NATHAN MARCUVITZ (S295) 295 

since Pn(x) = (— 1)'*P„( — x). By (3.8) it is evident that the normahzed eigen- 
functions are discrete and given by 

(4.7) 0, = (n + l/2y''P„ (cos 6), 

the orthogonality being with respect to the weight function sin d. 

4:Si2 . Oi < e < IT 

The characteristic Green's function is defined in the new domain by (4.1) 
but will be subject to the boundary conditions (d/dS) G{d, B') = at the regular 
point = di and finiteness at the singular point = t. As before ^m \ 9^ 
with X = v{v -jr !)• 

The relevant solutions of the homogeneous equation are in this case 

T{0) = P,(cos 0)-f-PX-Gos 0,) - P,(-cos (9) — P,(cos (9i) 
au do 

(4.8) 

U{0) = P,(-cos 0) 

and hence by (3.16) with o^o = ^1 . 

G(e, B') 

(4.9) 

P,(cos 9<) -T^P,(-cos Si) - P,(-cos 9<) ^P,(cos e,) P,(-cos e>) 

The completeness relation is 

sm ^ 2Tn '^ 

with the contour extended about all the singularities of G in the \ plane. The 
latter are in the form of simple poles occurring at the roots Vi of 

(4.10) {d/d0)P,X- cos 610 = 

which in view of the positive definite Hermitean character of L lie along the 
positive real X axis. No poles lie at integral values of v in virtue of the vanishing 
of the numerator of (4.9) at these points. Evaluation of the above integral 
along the same contour as in Figure 4.1 leads by the residue theorem to the 
discrete representation 

(A in g(^ - n ^ V (^. + |)P...(-cosg)P.,(-cosO 

^ • ^ sin^' 4^sin..,rr d' ^, ^^ / d ^, r\ 



296 (S296) ELECTROMAGNETIC WAVES 

from which the normalized (with respect to the weight function sin B) eigen- 
functions are readily identified. 

4bi . 7? = \y q = —k^,w = r~^; r^ < r < Vz 

The characteristic Green's function will be defined by 

(4.12) lidydr') + A:' -f (\/r')]G{r, r') = - 5(r - r') 
with 

(4.13) k^ Si positive reai constant, and ^ G = 

at the regular boundary points r = Vi and r2 . The operator L is thus Hermitean 
and hence the restriction ^m\ 9^ assures a unique solution to (4.12). 

Solutions of the homogeneous operator equation possessing a vanishing 
derivative at 7\ and 7-2 are if X = —v(v-\-l) 

T(r) = jXkr)ni{kn) - nXkr)j'Xkr,) 
(4.14) 

U{t) = j\{krK(kr,) - 7iXkr)j:{kr,) 

where 

(4.15) jXx) = (7rx/2)^/V.,.(x) and n^x) = {'Kx/2y^'N,,.{x) 

are the spherical half order Bessel and Neumann functions^^ possessing a 
Wronskian j,(x)n^((i) — n^{x)ji{x) = 1. The Wronskian of the solutions (4.14) is 

W^U, T) = k[fXl^r,)K{kT,) - ni{kn)ji{kr,)] 

and hence bj^ equation (3.15) 

^, ,. ^ lj.{kr<)n:ikn) - nXkr,)j:{kr,)][jXkr,)7i:{kr,) - nXkr)j:(kr,)] 
^''' "^ ^ klfXkrOnXkn) - 7i:{kr,)j:{kr,)] 

where, as above, r< and r> denote the lesser or greater, respectively, of the values 
r and r'. In vicAV of the regularity of the functions T and U in the X plane, the 
only singularities of the characteristic Green's function correspond to the zeros 
of the denominator in (4.16). The Hermitean and non-definite nature of the 
operator L imply that these singularities are located on both the positive and 
negative real X axis and have the form of simple poles. In the vicinitj- of the 
typical pole X, = —v^(vi + 1) 



(4.17) W.{U, T)== - ±W, ir-T^ + ■ 



Id" 'l<2v, + 1 



^^This notation differs by a factor of x from the corresponding notation in Stratton, J. A., 
'Electromagnetic Theory", (1941), Sec. 7.4; the former appears most convenient in vector 



problems, the latter in scalar problems. 



NATHAN MARCUVITZ 



(S297) 297 
The completeness relation (3.8) is given by 

r''8{r - rO = - ;r^. f G{r, r'; X) d\ 
with the contour as shown in Figure 4.2 extended about all the singularities of G. 



€ 



» 



Figure 4.2 



Evaluation of the residues at the poles then yields 
r''8{r - /) = Z (2^i + 1) 

(4.18) 

[iv,(fey)^'.(^n) - n,Xkr)jiXkri)][j,Xkr')KXkn) - n,Xkr)jiXkn)] 

[{d/dv)WA.. 

from which the spectrum is manifestly discrete with eigenf unctions 

(4.19a) { [(a%)VV j ''^^'"^^''^''"^^'^^ ~ ri.Xkr)j:Xkr,)] 

normalized to unity with respect to the weight function 7^~^, and with eigen- 
values X, = —Vi (v, + 1) given by 

(4.19b) W^XU, T) = 

4b2 . < r < r2 

The characteristic Green's function will be defined in the domain < r < r^ 
as in equation (4.12). Let \ = —v{v-{-l) or more definitely v -{- ^ = 
— i(X — iy^^, and choose 



(4.20) 



(Re (v+ i) = 6m (X - i)'"' > 0. 



The singular point at r = is then of the 'limit point" type and hence for a 
unique characteristic Green's function the boundary conditions may be given 
in the form 



(4.21) 



G finite at r = 

dG/dr = at r = r2 . 



298 (S298) 



ELECTROMAGNETIC WAVES 



The desired homogeneous solutions (3.12) are 

(4.22) Uir) = jXkrKikr,) - nXl^mkr^mr) = j^kr) 

and possess a Wronskian 

W,iU, T) = k/Xkr,). 

The completeness relation (3.8) then follows from the resulting characteristic 
Green's function (3.15) as 



(4 23) r"5(r - r') = ~4 J^^^^<'>^J^^^^^>^^'^^^''^^^ ~ ^>(^'^)i>(AT2)] 

27r2 ^ kj'Jkr,) 



d\ 



with the contour extended about all the singularities of the integrand in the X 
plane. Since the operator L is stiU Hermitean for this case, all singularities he 
on the real X axis. There may be a discrete number of singularities in the form 
of simple poles located on the real axis X < i at the zeros oi j',{kro). if any; and 
there is also a continuous singularity in the form of a branch cut extending along 
the real axis from X = i to + oo . The latter delimits the positive imaginars' 
branch of (X — J)^^^ on which jv{kr) 'j^ikro) is regular. The required contour 
for the integral in (4.23) is shown in Figure 4.3. On e\-aluating the residues at 



€ 



> 



Figure 4.3 



the poles, and expressing the branch cut integral in terms of v, one obtains for 
equation (4.23) 



'"Kr - n = Z (2^. + 1) 



j\Xkr)j.Xkr') 



kl{d/dv)j:(kr^/n:(kr^], 



(4.24) 



+ 






where c = —h-\-0 and the Vi are determined by 
(4.25) j:,..(At,) = 0. 

It is to be noted that (4.25) has a finite number of solutions Vi for At., > tt but 
finite. In view of the symmetry- in »/ + J of the path of integration, the integral 



dGldr = 


Sitr = Ti 


dGldr - ikG -^ 


at r — >oo 



NATHAN MARCUVITZ (S299) 299 

over V in (4.24) is symmetrical in r and r^^* From the representation in (4.24) 
it is evident that the spectrum possesses in general both a discrete and a con- 
tinuous part. The non-diagonal (biorthogonal) integral operator in (4.24) can 
be readily cast into a diagonal form (cf. equations 3.21a and b) from which 
both the discrete and continuous eigenf unctions are directly obtained; however, 
the representation in (4.24) indicating the continuous eigenfunctions in bi- 
orthogonal form is frequently more useful. 

4b3 . ri < r < CO 

A characteristic Green's function in the interval Vi < r < <» will again be 
defined as in equation (4.12) but subject to the boundary conditions 



(4.26) 



The latter so called '^radiation" condition with k real may be phrased alterna- 
tively as the condition G ^' at r —^oo H ^m k > 0. In view of the complex 
nature of the latter condition it is evident that the operator is non-Hermitean 
and hence to assure a unique G, X must be suitably restricted. For k positive 
real a simple evaluation of the location of the eigenvalues via Green's theorem 
shows that the desired restriction is ^m X > 0. Moreover, it is of interest to 
note that although in the Hermitean case the singular point at infinity is of the 
''limit circle" type, for the non-Hermitean case with complex k it becomes a 
''limit point" case. 

The homogeneous solutions satisfying the boundary conditions (4.26) are, 
respectively, ii\ = —v(p + 1) 

T(r) = j y{kr)n'Xkri) — nXkr)ji{kri) 
(4.27) 

Uir) = hl'\kr) 

where in accordance with the previous definitions (4.15) 

hl'\x) = {7rx/2y''Hll\{x) 

defines the spherical Hankel function. The Wronskian of the two solutions 

(4.27) is 

(4.28) WXU, T) = -khi'''(kr,). 
The completeness relation follows by (3.8) and (3.15) as 

(4 29) r^'8(r - r') = -^ 4 lUkr^nJikn) - nMr)j:{krM'\kr,) ^^ 

27r^•^ khl'^'ikrO 



"Note that n,{x) = [cos {u + h)ir j,{x) — j^,(x)]/sm {v + ^)7r where x = kr'. 



300 (S300) 



ELECTROMAGNETIC WAVES 



the contour being extended about all the singularities of the integrand in the 
X plane. Since T{r) and hl^\kr)/hl^^' (kr^) are integral functions of X, the only 
singularities occur at the roots v, of 



(4.30) 



KV\kn) = 



and have the form of simple poles. The relevant roots are tabulated in Bremmer 
(loc. cit.) p. 44 and are found to occur in the lower half plane ^m X < or in 
the first quadrant of the jz-plane. The contour of integration for (4.29) is some- 
what as sho\Mi in Figure 4.4 




Figure 4.4 
On evaluation of the residues at the indicated poles there is obtained for (4.29) 

hl]\kr)K]\kr') 



(4.31) 



''d{r - r') = 2: (2... + 1) 



ik[{d/dv)hl'''{kn)/rxkn)l, 



where the sum is to be taken over all the roots v^ of (4.30). The spectrum is 
evidently discrete. The eigenfunctions normalized with respect to the weight 
function r"" are 



(4.32) 



2i^v + 1 



.ik[{d/dv)K'''{kn)/j'Xkr,)l, 



h^^ikr). 



It is to be emphasized that in view of the non-Hermitean nature of the operator 
L, orthogonality is to be understood in the symmetric and not in the Hermitean 
(complex conjugate) sense. 

4b4 . < r < 00 

The characteristic Green's function is defined in equation (4.12). As in 
examples (b2) and (bg) let us choose (Re {v -\- \) = ^m (X — i)^''" > and 
^m k > 0. It is thereby implied that the singular points r = and ^ are of 
the 'limit point" type and hence requirements of finiteness at these points 
suffice to uniquely characterize the G(r, r') of equation (4.12). 

The required homogeneous solutions are 

nr) = j,{kr) 
(4.33) 

U(j-) = h':\kr) 



NATHAN MARCUVITZ 



(S301) 301 



whose Wronskian is —ik. By (3.15) the expHcit characteristic Green's function 
becomes 



(4.34) 



G{r, r') = 



— ik 



it possesses a branch point at X = | and is regular in that branch of the Riemann 
surface of (X — lY^^ for which ^m (X — jY^^ > 0, i.e. excluding a branch cut 
along the real X axis from X = i to oo. On forming the contour integral of 
G(r, r') about the branch cut (cf. Figure 4.5), one obtains by equation (3.8) 
the spectral representation 



^ 



uuuuuuuuuuvuv 



Figure 4.5 



of the identity operator as 

(4.35a) r'^b{r - r') = 

(4.35b) 



hi 



jXkr<)K'\kr,) 
k 



d\ 



iic CJ'^''^"''^"'"^^^ 



V + 1) dv 



withe = -i + 0. 

The spectrum for this case is manifestly continuous. The non-diagonal, or 
biorthogonal, form^ of the completeness relation in (4.34) is usually most 
convenient as the basis for a transform theorem. It is of interest to point out 
the form of completeness relation deducible from the static (i.e. k^ — > 0) char- 
acteristic Green's function. On substitution of the small argument asymptotic 
relations 



(4.36) 






r(. + f) \2/ 

ij^r' (A- 

h)TT(h - v) \2l 



^2± 

sin {v + i)7rr(i 



i^cf. N. N. Lebedev, Dokl. Acad. Nauk. USSR, (1947), Vol. 58, No. 6. 
i^The symmetrical form with eigenfunctions proportional to /i / ^ ^ {kr) is obtained as in the 
transition from equations (3.21a) to (3.21b). 



302 (S302) ELECTROMAGNETIC WAVES 

and the identity 

cos^TT = r(i + ^)r(i - v) 

into (4.35b), there is obtained in the hmit /b^ — > 
(4.37) r''8{r - r') = -i-. ^^^ r'^V dv 

which is, of course, the basis for the Melhn transform. 



4c.. P = ~, 



q = -e, 



w = 



e{r)r' 



ri < r < r^ 



A characteristic Green's function is defined in the interval r^ < r < r^ by 

and subject to the boundary conditions dG/dr = at the regular points r = r-i 
and fs . The parameter k^ is a positive real constant and 



= ei 



<r) 



ri < r <T2 
r2 < r < Vs 



with €1 and €2 real constants. It follows from (4.38) that both (l/eir))/ (dG/dr) 
and G are continuous at all points r 9^ r' . In view of the realit}^ of the operator 
L the restriction ^m X 7^ assures a unique G{r, r'). 

The discontinuous nature of e(r) implies a like character for the solutions 
of the homogeneous form of equation (4.38). Thus if X = —v{v-r\), the 
solution t{r, rg) with vanishing derivative at r = ri , wdth (1/e) (dt/dr) and f 
continuous at r = ^2 , and normalized to unity at r = rj is piecewise represented 
by 



(4.39) 






The index a denotes 1 or 2 depending on whether ri < r < ra or r^ < r < 7-3 , 
ka = kieaY^^, and in accordance with the format in equations (3.17-8) and 
the definitions (4.15) 



(4.39a) 



c.(x, y) = jXx)ni{y) - nXx)fM 
s.{x, y) = 3Ay)nXx) - nXy)jXx) 



x: = 



€1 S„(/Ci7'i , /Cifa) 



NATHAN MARCUVITZ (S303) 303 

Correspondingly, the solution u{r, ra), normalized to unity at r = ra and with 
vanishing derivative at r = r^ , is given by 

(4.40) u{r, n) = cXKr, kj,) + X^ ''^Yjef""^ 

€2 Sv\K2^Z J "^2^2 J 

In view of equations (4.39-40) the characteristic Green's function has the dis- 
continuous representation (cf. 3.19) 

G{r,r') 
(4.41) 

where 

(4.41a) X, = Z; + X: . 

The meromorphic character of t(r, ra) and u{r, rs) implies that the only 
singularities of G{r, r') in the X plane lie at the zeros Vi of X;, . The Hermitean 
and non-definite character of the operator L imply that these singularities are 
located on both the positive and negative real X axis and in fact have the form 
of simple poles. As in equation (3.20) the contour integral of the characteristic 
Green's function along a path like that shown in Figure 4.2 leads on evaluation 
of residues to the completeness relation 

Kr - r') _ ^ {2v, -f- 1) \^ (T, r l^ T^ - Y- ^.JMiJmA] 
l/e(r)r ,, (dX,/d»'),.. L /Ca/€« J 

(4.42) 

The eigenfunctions normalized to unity with respect to a weight function 
€(r)~V~^ thus have the discontinuous representation 



r 2v, + 1 1 
l{dXJdv).J 



''\cyXhr, hr,) r,<r<r, 



CuXkir2 , k,rO ' 
(4.43) 

[2.^+1 1-^- c.,{k.r, te) , <,<, 

l{dXJdv)yJ c.Xk^r. , hr,) ' ^^ ^ ^ ^ ^3 , 

with Vi determined by 

X,, = 0. 

By comparison of (4.43) with the eigenfunctions (4.19a) of example bi it is 



304 (S304) 



ELECTROMAGNETIC WAVES 



evident that the latter are just the special case €i 
former in a somewhat different form. 



= 1 and 7-2 = rg of the 



4c2 . < r < /'3 

The characteristic Green's function is defined as in (4.38) but with 
j\ = 0. Since ?' = is a singular point of the "limit point" variety (pro\'ided 
(Re {v -{- h) > 0), the results of the previous example c^ are directly applicable 
on letting r^ — > 0. The nature of the spectrum, however, becomes more apparent 
on direct investigation of the present case. 

With the same choice of parameters as in Ci but with (Re (v -\- i) = 
^m (X — \y^''^ > the desired homogeneous solutions t{r, r2) and u{r, rg), the 
former finite at r = and the latter with vanishing derivative at r = r^ , are 
the same as in equations (4.39) and (4.40) with the sole modification 



(4.44) 

which implies that for r < r. 



x; = 



t{r, n) 






The characteristic Green's function possesses the same discontinuous repre- 
sentation given in (4.41) but with X~ given by (4.44). In view of the branch 
point singularity of X"^ at X = J the solution t{r, r.2) is no longer a meromorphic 
function of X. Hence the singularities of the characteristic Green's function in 
the X plane are not only simple poles located at the zeros of X, (cf. equation 
(4.41a)) but also a branch cut along the real X axis from X = J to 00. The 
associated spectrum of the operator L for this case thus comprises not only a 
discrete but also a continuous set of eigenvalues; it is the continuous set that is 
not explicitly e^ddent on taking the limit i\ — > of the results in Ci . 

The completeness relation (3.8) is obtained bj^ integration of the character- 
istic Green's function along a contour in the X plane similar to that sho^n in 
Figure 4.3. As in example bs the integral around the poles is expressed in terms 
of residues; however, the branch cut integral is expressed in terms of the variable 
V. The completeness relation thus takes the form 



eir'Y'bir - r') 



{2v, H- 1) 

(ax./a.),, 



CyiikaT, kaTo) 



X 



ka/^a 



] 



(4.45) 



+ 



1 r^'^^l n 7 ^ ^- BXkaT.kj.f \ 



• cXkal 



k.r,) + X 



+ s,{ky, kar.2) 

ka/^a 



]^ 



NATHAN MARCUVITZ 



(S305) 305 



with c = — I + 0. The symmetry in r and r' of the integrand in (4.45) is a 
consequence of the evenness mv -\- \oi the functions c^ and s^ (cf. equation 3.32). 
The discrete eigenfunctions, normahzed to unity with respect to the weight 
function l/e(r)r^, follow from (4.45) as 



(4.46) 



with the Vi determined by 






2r3)_ 

n) 



<r <r. 



To < r < To 



The continuous eigenfunctions are displayed in biorthogonal form in the v in- 
tegral of (4.45); the eigenfunctions in symmetrical form are proportional to 
u{r, Ti) as is evident from the corresponding form in equation (3.21b). The 
above results should be compared with those for the special case ei = €2 = 1 
and r2 = r^ discussed in example b2 . 



4c, 



< r < 00 



The characteristic Green's function will be defined by equation (4.38) but 
subject to the boundary conditions 



f = 

ar 



at r 



G^O 



as r 



The parameter k^ is now chosen as complex with a positive imaginary part 
(however small) and 



:(r) 



^2 



fi < r < r2 



r2 < r. 



The operator L is non-Hermit ean in this case and hence ^m \ 9^ does not 
suffice to insure a unique G{r, r'); rather, as in example bs , it is necessary to 
restrict X to ^m X > 0. In view of the ''limit point" character of the singular 
point at c» when k^ is complex, it is not possible to consider this case as the 
limiting form rg — > 00 of case Ci in which, since /:^ is real, the point at infinit}'- is 
of the 'limit circle" type. 

Solutions of the homogeneous equation (4.38) possessing the requisite con- 
tinuity properties at r = rg and satisfying the boundary conditions at r = r^ 



306 (S306) ELECTROMAGNETIC WAVES 

and 00 are given, respectively, by equations (4.39) and (4.40; except that in 
this case 

(4 47'\ ' X'^ = — ("^2^2) 

which imphes that for r > rg 

u(r, r,) = -^i!!M. . 

The characteristic Green's function is discontinuously represented as in equation 
(4.41). Since 

H[lL,(x) = exp {-i(v + i)T}Hi'U,ix) 

it follows that X^^ is an even function of v, and hence an integral or meromorphic 
function of X, as is X~ . Thus the onl}^ singularities of the characteristic Green's 
function are simple poles located in the X plane at the zeros of X, . The location 
of these poles is for the most part that indicated in example bs ; however, in 
addition to the poles located in the fourth quadrant of the X plane, there may 
occur poles on the real X axis Avhen ci e, > 1. 

On integration of the characteristic Green's function about a contour (cf. 
Figure 4.4) enclosing all its singularities, one obtains the completeness relation 
(3.8) in the form given in equation (4.42). The spectrum is evidently discrete. 
The eigenf unctions normalized to imitj^ with respect to a weight function 
l/e(r)r^ become in this case 






r, < r < r. 



r, < r 



c,Xkir2 , k,i\) ' 
(4.48) 

r 2v, + 1 1^-^^ hl^'ik^ 

l{dX^/dv),J hl]\k,n) ' 

where the Vi are determined from the roots of 

e, s'^{k,i\ , k,u) €2/ii^^(Av'2) 

As in the special case ei = €0 = 1, ^i = fo treated in example h^ orthogonality 
is to be understood in the sj^mmetrical sense despite the complex natiue of the 
eigenfunctions. 

4c4 . < r < 00 

The characteristic Green's function is defined b}^ equation (4.38) and subject 
to boundary conditions 

G finite as r -^0 

G zero as r -^00 



NATHAN MARCUVITZ (S307) 307 

with the parameters A;^ and e(r) defined as in the previous example Cg except 
that n = 0. The restrictions (Re (v + i) = ^m (X - i)'^' > and ^m /c' > 0, 
respectively, insure that the singular points r = and r = co are of the ''limit 
point" type. The further restriction ^m X > on the branch ^m (X — lY^^ > 
is necessary to insure a unique G{r, r') for this non-Hermitean case. 

The homogeneous solutions t{r^ T2) and w(r, Vo) satisfying the continuity 
conditions at rg and boundary conditions at r = and <» are again given by 
equations (4.39) and (4.40). The reactances X~ and X\ at r2 are the same as 
in equations (4.44) and (4.47), respectively. The characteristic Green's function 
is of the form shown in equation (4.41). As noted above X~ has a branch point 
at X = i whereas X\ is regular. Hence G(r, r') possesses singularities in the 
X plane in the form of a branch cut along the X axis from X = } to 00 and simple 
poles at the complex zeros ^^ , if any, of Xy . The spectrum for this case thus 
may be both discrete and continuous. 

Evaluation of the completeness relation (3.8) requires the integration of 
the characteristic Green's function along a contour enclosing both the poles 
and the branch cut of the latter. The result may be given in the form shown in 
equation (4.45) from which on insertion of the relevant expressions for X~ and 
XX the discrete eigenfunctions are seen to be 

lidX,/di'),J h.(kin) 
(4.49) 

with the Vi defined as the roots of 

Y 1^ Jv\k'lf2) I ^2 i^v (^2^2} pw 

The discontinuously represented and biorthogonal eigenfunctions of the con- 
tinuous spectrum likewise follow from (4.45) as 



(4.50) 









ihr) 



(/Cs^a) ' 



< r < r. 



r.2 < r, 



where +0 — ^oo <j,-f-i^'<-|-0 + ioo defines the continuous spectrum (note 
that a l/(27^^) factor is understood in superposing eigenfunctions). An alterna- 
tive symmetric orthogonal form for the eigenfunctions of the continuous spectrum 
is also obtainable as in equation (3.21b); in this case the eigenfunctions are 
proportional to u(r, r-i). 



308 (S308) ELECTROMAGNETIC WAVES 

5. Field of a Point Source. Multi-dimensional Green s Functions 

As noted in Section 2 the determination of the electromagnetic field pro- 
duced by an arbitrary vector current distribution may be reduced to two scalar 
problems for the potentials. The latter are defined by inhomogeneous partial 
differential equations which in turn may be reduced to ordinary one dimensional 
differential equations on introduction of an appropriate set of eigenfunctions, 
or modes. Since the field of a known current distribution follows by integration 
from the corresponding field of a point or dipole source it is sufficient to treat 
only the latter problem (cf. Section 3b), i.e., if we exclude diffraction problems 
in which the current distribution is not known and also has to be determined. 
For definiteness the discussion below will refer to the field of a vertical (radial) 
electrical dipole, which excites only E type modes. The H type field of a vertical 
magnetic dipole follows by duality; however, the field of an arbitrarily oriented 
electric or magnetic dipole entails a somewhat greater degree of complexit}^ 
because of the excitation of both E and H type modes, but nevertheless follows 
by similar methods (cf. Section 2). The nature of the spherically stratified 
region in which, for illustration, it is desired to obtain a representation of the 
field is shown in Figure 5.1. The examples to be discussed are distinguished 




Figure 5.1 

by the values ascribed to the radii i\ , r., , r^ of the boundaries (if an}') and to 
the relative dielectric constants ci and e. (for simpHcity, /ii = )U2 = 1) of the 
strata. 

The various representations of a dipole field are clearly exhibited by phrasing 
the relevant mathematical problem as a Green's function problem. The choice 
of a vertical dipole implies that only two dimensional problems will be considered. 
The Green's function of interest then satisfies the inhomogeneous equation^' 



with boundary conditions and variability of e and fi to be stated. Equation (o.l) 



5(r - rO • ^/ 
^ sm ^ 



^^Strictly stated, the azimuthal symmetry of the field of a single vertical (radial) dipole 
obtains only for 6' = 0. To emphasize the symmetry of the mathematical problem 0' will be 
retained explicitly; this corresponds to a ring of radial electric dipoles. 



I 



NATHAN MARCUVITZ 



(S309) 309 



is evidently a special case of equation (2.25b). As discussed in Section 2, 
equation (5.1) represents but one of a variety of ways of phrasing the field 
problem for a vertical electric dipole. One virtue of the Green's function 
formulation is that there exists a formal prescription for solving equation (5.1) 
in terms of the eigenf unctions or characteristic Green's function of the com- 
ponent one dimensional operators in (5.1). Thus, as discussed in Sections 3 
and 4, let the eigenfunctions, eigenvalues, and characteristic Green's functions 
of the two relevant operators in (5.1) be defined by 



(5.2a) 

(5.2b) 

and 

(5.2c) 

(5.2d) 



(A 



^^smej-^ + X^sinOkm 



^)iii 







Jsin^^+Xsin^ 



)G{d, d^;X) = 



8{d 



(l7l+^^'' + ^>'W = o 



\dr e 



|+fcV + |.)(?(r,r';X) 



5(r - r'). 



The boundary conditions on G and G correspond to those on g (r, r') at the end- 
points of the 6 and r intervals, respectively; the conditions on }i and </)i are 
given in terms of those on G and G by relations of the form (3.3b). In terms of 
the above defined functions the completeness relation (3.8) is given in the 9 
interval by 



8(6 - $') 
sin d' 



(5.3a) ""^^e^ "^ ~^L ^^^' ^'; ^) ^^ = ^ l(»)Ue'), 

and in the r interval by 

(5.3b) er"S(r - r') = - ^. f^ G(r, r'; \) dk = E ^.(r)^,^'), 

where the contour C is extended about all the singularities of G in the complex \ 
plane, and where C encloses all the singularities of G in the X plane. With the 
knowledge of equations (5.2-3) the solution to equation (5.1) may be represented 
immediately in the alternative forms 



(5.4a) 



9(r,r0 = T.G{r,r';-\)Ue)Ue') 



(5.4b) 



--i^jG(r,r''^-me,e'',i) 



d\ 



310 (S3 10) ELECTROMAGNETIC WAVES 

(5.4c) = - ^-^ J G{e, d'; -\)G{r, r'; X) d\ 

(5.4d) = E Gie, e'; -X,)<^,(r)0,(r') '' 

of which the first two are so called representations in d, the latter two in r. 
Equations (5.4) may be verified by noting that the operation 

111 , , 2 , i_ _J_ ± - .A_ 

under the sum (integral) sign of each of the right hand members in (5.4) yields 
the desired result: — 5(r — r')b{d — 6') /mi 6'. The interchange of summation 
(integration) and differentiation is permitted in view of the identity' of the 
boundary conditions on Q (r, r') and on G{r, r') and G{6, 6'). 

Equations (5.4a-d) provide four different representations of the Green's 
function defined in (5.1), each representation having different convergence prop- 
erties depending on the range involved. For example: in far field calculations 
of plane wave scattering from small (compared to a wavelength) sphericalh^ 
stratified structures equation (5.4a) is most convergent; in far field calculations 
when the wavelength is small — the geometrical optical limit — equations (5.4b) 
or (5.4c) are most useful; in the calculation of the far surface field of a dipole 
located near the surface of a large spherical structure the representation (5.4d) 
is most convergent. From any one of the representations in (5.4) the others 
of (5.4) may be derived by appropriate sum — integral equivalences or contour 
deformations (cf. Section 3). Thus, Watson's classical treatment of the spherical 
earth and allied problems entails a transition from the representation in (5.4a) 
to that in (5.4d) via the intervening representations. The desirability of se- 
lecting the relevant representation without intervening function theoretic con- 
siderations should be evident. 

The characteristic one dimensional representations obtained in Section 4 
will now be employed to obtain alternative representations of the vertical 
electric dipole field in the stratified region i\ < r < rg , < ^ < it, sho^^Ti in 
Figure 5.1. Thus, we seek a solution of equation (5.1) subject to conditions of 
finiteness at ^ = 0, tt and to various, as yet unstated, boundary- conditions at 
r = ri , fs . In view of the representations (4.4) and (4.6) in 6, and of the repre- 
sentations (4.41) and (4.42) in r, the solutions (5.4) may be written more ex- 
plicitly in the notation of Sec. 4c as 

(5.5a) q{r, r') = Z ^ ^"'"^ ' '^-'•^^'(^'■'•> ' ^'''^^ (n + |)p„(cos e)P„(eos 9') 



(5-5'') = 2^- / 



i^See footnote 8. 



ty{kar^ , hgT^ uXkaT^ , hj-^) Pyjcos ^<)P,(— cos ^>) ^ 
-Xy ' (2/7r) sin t/TT 



NATHAN MARCUVITZ (S311) 311 

^^■^""^ ~2TriJc -(2/7r)sin^x * -X, "^^ 

^ -(2/7r)sin^,7r (^X,/a^),, 

(5.5d) 't,t{kar, A:„r2) U^{ky, kaV^ 



+ 



where 



J_ r^'-'^ PXcos (9<) P,(-cos ^>) tXkaV, k^r^) uXkaV', k^r^) 
2'jri jL^io= (2/7r) sin ^tt X^ 

'{2v + 1) 6/^ 
^Xx, y) = c,(x, 2/) - X- ^^^ 

^^.(^, 2/) = Cv{x, y) + X: ^-^^ 

rial ^a 

and where the index a is 1 or 2 depending, respectively, on whether r (or r') is in 
the range i\ to rg in which e^ is ei or the range rg to fg in which €« equals €2 . 
It should be noted that since X=— z^(z^+l) = — X, the contours C and C are 
simply related when drawn in the same X, or X, or v plane. Hence the representa- 
tions (5.5b) and (5.5c) differ only slightly from one another; the transition 
between these two representations involves, however, certain function theoretic 
properties of the integrand which need not be adduced in the direct application 
of the representation theorems (5.5b) or (5.5c). It is evident that the form 
(5.5d) corresponds in general to the existence of both a discrete and continuous 
set of eigenfunctions in r. 

In the following we shall confine ourselves to a more detailed discussion of 
the representation (5.5d) which expresses the vertical dipole field in terms of 
waves guided along the ^-direction. It is this form which is most convergent in 
many spherical propagation problems. The characteristic representations de- 
veloped in Sections 4b and c provide explicit values for X~^ and X\ in a number 
of stratified regions. Since it is of more interest to indicate the detailed nature 
of the spectrum rather than the detailed structure of the field in these cases, the 
latter will be omitted in the following. Detailed discussions of field structure 
for e(r) variations encountered in practice are contained in the works of Bremmer 
(loc. cit., Chaps. VIII to X) and Booker and Walkinshaw (loc. cit.). In parallel 
with the discussion in Sections 4b and c the following cases are considered in 
more detail: 

(1) ag/ar = ^ at r = ri and n 

This case is of interest in connection with an idealized tropospheric layer of 
constant dielectric constant ci extending from ri to r., and an isotropic ionospheric 



312 (S3 12) ELECTROMAGNETIC WAVES 

laj^er of constant dielectric constant €2 from rg to Vo, ; these layers are bounded 
below by a perfectly conducting earth at ri and above by a perfectly conducting 
sheath at r^ . The nature of the propagation is dependent on the relative values 
of €1 and 62 . 

Since X^ is a meromorphic function in this case, there is only a discrete 
spectrum. On insertion of the values for X~ and XT from equations (4.39a) 

Y 

and (4.40a), equation (5.5d) becomes for ri < , < r2 and < ^ < tt^^' = 0) 

(^ n o( '\ = ^ P., (-cos e) {2v, + 1) c^Xhr, k,rO c^,Qz{r' , k,r,) 
i^.oj W, ? ; Z. _(2/^) gin ^^^ {dXJdv),, c^XKn , k,r,) c^Xhr^ , hn) 

with Vi determined by X^. = X~ -\- X,t = 0. For large kr the latter equation 
can be approximated by a saddle point or by a W.K.B. procedure as 

(5.7) - sin /3i tan [7i(r,) - 71(^1)] + - sm /S^ tan [72(r3) - 72(^2)] = 
where 

(5.8) cos/3. =^^^ = 1^ 



ya{r) = kj- (sin ^^ - /3« cos /?«) + 



with a = 1 or 2. If /i«/r2 « 1, where h^ = r^ — i\ and /ia = rs - r2 , equation 
(5.7) can be replaced by the ''parallel plane" approximation 

[{k\r - r-)^'7ei]tan(/c^ - f;)''%, 
(5.9) 

+ [{k\2 - r?)''7^2] tan {k\2 - ^r% = 

provided A;^ = /c(e„)^^^ 7^ Ti • The zeros of (5.9) can be readily found and, 
when ea is real, they yield for X, = —Viivi + 1) real values of two distinct types: 
those for which X^ - i > 0, i.e. Vi + i imaginary, and those for which X, - 
I < 0, i.e. Vi + i real. The former values characterize modes that attenuate 
in the ^-direction, ^^ whereas the latter correspond to modes propagating in the 
0-direction. There are only a finite number of propagating modes (for finite 
n — ^1) and these can be further distinguished by whether ^, lies between 
k{e,y^^ and /c(e2)'^' or by whether fi is less than both k{e,y^^ and kie-^)'''. The 
modes for which A;(e2)'^^ < Ti < ^(^1)'^" are so called ''surface waves" or "trapped 
waves." They correspond to waves having no r propagation (attenuated) in 



^^Xote that for Pi large and with a positive imaginary part, and for 1/vi < d < tt — 1/vi 

P., (-cos^) ^ (iriY" exp {».- + ^)^i 
-(2/7r) sin ViTT \2 / [{vi + 1) sin 6]''' * 



NATHAN MARCUVITZ (S3 13) 313 

the dielectric €2 and suffering internal reflection in dielectric ei — or conversely 
if 62 > ei . 

The non-Hermitean case with complex e^ yields complex values for X, , but 
save for the additional attenuation introduced thereby this case is for the most 
part the same as for real e„ . In the limit case e^ = €2 there are no surface 
waves 

(2) dQ/dr = atn ,Q finite at r, = 0. 

This case sheds an interesting light on the effect of a spherical earth of 
dielectric constant €1 and radius Vz upon propagation in the presence of a per- 
fectly conducting ionospheric sheath at r^ . 

On insertion into equation (5.5d) of the relevant values for X~ and XX 
from equations (4.44) and (4.40a), one obtains in virtue of the non-analytic 
character of X~ both a discrete and continuous spectrum for this case. The 
continuous modes correspond to the coalescence of the discrete modes with 
Xi — i > of the preceding example. On the other hand the discrete modes 
for this case are found for those values \i = —Viivi -{- I) < j that satisfy the 
resonance equation X^,. = X~i -j- X^^ = or, with the same approximation and 
notation as in equations (5.8-9), 

(5.10) - ^ sin /3i tan y,{r,) -f [{k\ - ^T'/e,] tan (k'e^ - ^T% = 0. 

€1 

Again only a finite number of such modes are admitted if kr^ is finite although 
of course this number may be large. Surface waves are possible both for €1 > €2 
and for €2 > ci , in the latter case an inverted surface wave appears. 

The non-Hermitean case with complex €« , as in the previous example, 
requires a slight modification of the above results in that all the Vi are complex 

(3) dQ/dr = at r, Q -^ as n -^°° 

The idealization involved in this case appears suitable for the description 

' of anomalous propagation in the presence of a tropospheric layer of dielectric 

constant ei and height hi = r2 — r^ bounded below by a perfectly conducting 

spherical earth and above by an atmosphere of dielectric constant €2 . It will 

be assumed that eg/ei < 1. 

The meromorphic reactances X~ and X"^ are given in this instance by 
equations (4.39a) and (4.47). The associated spectrum is discrete and in the 

T 

range r^ < > < Vz (with 6' = 0) the representation (5.5d) becomes the same 

as in equation (5.6) where the Vi are again the roots of X,,. = X~ -}- X,^ = 0. 
These roots are either complex or real (if e^ is real). With the same approxima- 
tions and notation as in equations (5.8-9) the complex ^. are determined b}^ 

(5.11) [{k\ - rv)''7€j tan {k'e, - ^^Y'X + (k^/e^) sin ^^ tan 72(r2) = 0. 



314 (S314) ELECTROMAGNETIC WAVES 

For numerical convenience one defines 

(5.12) V, = k,r, + {t^.y^'u 

To the indicated approximation 72(^2) ~ ^2^2 /3^/3 ~ [( — 2ri)^^^]/3 and hence 
equation (5.12) becomes 



m - ky/e,] tan (kl - klf'h, 
(5.14) 



h (-2r.-) ' 
€2 (^2^2) 



+ /7 xi/3 tan 



I + I (-2r..)'"] = 0, 



from which an infinite set of r^ , and hence Vi , can be found. The numbers r, so 
found are closely related to those encountered in the case of an imperfectly con- 
ducting earth (cf. equation (5.16) below). In the range where real roots Vi are 
possible tan 72(^2) ~ i; hence the equation X^. = has by (5.11) the ''parallel 
plane" approximation 

(5.15) l{k% - f^'-'V*.] tan (k'e, - t]r% - [(f, - k%r/.,] = 0. 

Equation (5.15) admits only real roots ^,- which are finite in number if khi is 
finite. They characterize "surface waves" or ''trapped modes" propagating in 
the 6 direction. The surface waves encountered in example 1 differ only slighth^ 
from those for this case, as is evident from the fact that in the relevant range 
of ^i the right hand tangent in equation (5.10) is approximately unity. This 
is indicative of the near independence of the surface waves on the nature of 
the boundary surface at fg — ^ oo . A feature in marked contrast with that of the 
waves, defined by (5.11), which are radically affected by the nature of the 
boundary at r^ . The latter fact is intimately related to the "limit circle" 
character of the singular point r^ in Hermitean case. 

As in the previous example the case of complex e^ introduces minor modi- 
fications. The limiting case €i = €2 likewise admits complex r, , as is seen from 
equation (5.14), but there are no surface waves. For ci = €2 the functions 
c^,(kir, ki)\)/c^,{kir2 , k^r^) in (5.11) should be replaced by hiWkir) /hlWki}\) . 

(4) g finite at r^ = 0, 9 -> rg -^00 

This is the oft treated problem of a spherical earth of dielectric constant 
ci and radius /'a bounded above by an atmosphere of dielectric constant €9(62 < ci). 

The reactances X"^ and XX are given in equations (4.44) and (4.47). The 
non-analyticity of X~ implies that the spectrum for this case is both discrete 
and continuous. The Vi for the discrete modes are determined as alwaj^s b}^ 
the zeros of the equation X,, = X~ -\- X^, = 0. The latter can be approxi- 
mated for large /c2r2 by replacing X~ by the left hand member of equation 
(5.10) and X^, by the right hand member of equation (5.11). On introduction 
of the parameters r, defined in (5.13) and use of the approximation tan 71 (r,) = i 
(i.e. assuming a slight amount of dissipation) this equation becomes 



NATHAN MARCUVITZ (S315) 315 

Equation (5.16), which is treated in detail by Bremmer (loc. cit. Eq. (21b) p. 
43), admits an infinite number of complex zeros r, . The associated v^ correspond 
to the modes defined by equation (5.12) of the previous problem. It is of in- 
terest to note that there is no discrete surface wave in this case, as is also true 
in the plane earth limit rg -^ <» , 

(5) Conical Region. d£/dd = at 6 = 6^ 

The mode representations just discussed are applicable to a variety of 
other cases. For example, if the spherically stratified regions of the preceding 
examples are bounded by a conducting wall at ^ = ^i , one has only to replace 
in (5.5d) the Green's function (4.4) by (4.9) to obtain the field of a radially 
directed, h-independent electric dipole distribution. The consequent change in 
the nature of the ^-propagation represents the sole modification in the results 
discussed above. 



Bibliography 

1. Bremmer, H., Terrestrial Radio Waves, Elsevier Publishing Company, New York-Houston, 

1949, Chapters II-VI. 

2. Weyl, H,, Uber gewohnliche Differentialgleichungen mil Singularitdten und die zugehorigen 

Entwicklungen willkurlicher Funktionen, Mathematische Annalen, Volume 68, 1910, 
p. 220. 

3. Titchmarsh, E. C, Eigenfunction Expansions, Oxford Univ. Press, 1946, Chapter II. 

4. Sommerfeld, A., Electromagnetische Schwingungen, in Frank, P., and von Mises, R., Die 

Differential und Integralgleichungen der Mechanik und Physik, Volume II, Rosenberg, 
New York, 1943, p. 891. 

5. Booker, H. G., and Walkinshaw, W., The mode theory of tropospheric refraction and its relation 

to wave-guides and diff ration. Meteorological Factors in Radio Wave Propagation. 
Report of a conference held on April 8, 1946, by the Physical Society and The 
Royal Meteorological Society. The Physical Society, London, 1946, p. 80. 

6. Friedrichs, K. O., Lectures on spectral representations of linear operators. New York Uni- 

versity, New York, 1948. 

7. Stratton, J. A., Electromagnetic Theory, McGraw-Hill, New York, 1941. 

8. Lebedev, N. N., On the representation of a function as an integral of a MacDonald function, 

Dokl. Acad. Nauk. U. S. S. R., Volume 58, No. 6, 1947, p. 1007. 



Si> 



Propagation in a Non-homogeneous Atmosphere* 

By BERNARD FRIEDMAN 
Washington Square College of Arts and Science, New York University 

I. Introduction 

The problem of explaining the propagation of radio waves around the sur- 
face of the earth has long been of great interest to both physicists and mathe- 
maticians because of its practical importance and because of its considerable 
mathematical difficulties. 

There are two fundamental approaches to the theory of such propagation. 
The first approach, which is that of Watson [1], van der Pol and Bremmer [2], 
and Rydbeck [3], and which has been applied mostly to ionospheric problems, 
assumes the earth to be a perfect sphere with finite or infinite conductivity, and 
expresses the field due to a radiating dipole as an infinite series involving spherical 
Bessel functions. Since the series converges much too slowly for practical pur- 
poses, it is transformed into a contour integral which can be evaluated by the 
method of residues. To find the residues of this integral, highly complicated 
asymptotic expansions for the spherical Bessel functions must be used. These 
asymptotic expansions, which are derived by the method of stationary phase, 
involve the Hankel functions of order 1/3 or the Airy integral, both of which 
occur frequently in diffraction theory. 

The second approach to the problem is that used by Pryce [4] and Booker 
[5] in England, and by Furry [6] and Pekeris [7] in the United States while 
studying the effect of the troposphere on radio wave propagation. In this 
approach, the earth is assumed to be flat but, following a suggestion of Schelling 
and Burrows [8], the index of refraction is modified so as to take into account 
the curvature of the earth. The field due to a radiating dipole is now expressed 
as an infinite integral involving ordinary Bessel functions. This integral, also, 
can be evaluated by the method of residues but it was realized that the residues 
are really the successive terms in an eigenfunction expansion for the field of the 

Paper presented at the June, 1950, Symposium on the Theory of Electromagnetic Waves, under 
the sponsorship of the Washington Square College of Arts and Science and the Institute for 
Mathematics and Mechanics of New York University and the Geophysical Research Direc- 
torate of the Air Force Cambridge Research Laboratories. 

*This work was performed at Washington Square College of Arts and Science, New York 
University and was supported in part by Contract No. AF-19(122)-42 with the U.S. Air Force 
through sponsorship of the Geophysical Research Directorate of the Air Force Cambridge 
Research Laboratories. 

317 (S317) 



318 (S318) * ELECTROMAGNETIC WAVES 

dipole. The propagation problem is therefore reduced to the study of an eigen- 
value problem for an ordinary differential equation. In case the atmosphere 
above the earth is assumed to have a constant refractive index, the eigenfunction 
expansion is in terms of Hankel functions of order 1/3, exactly the same func- 
tions which appear in the first approach. 

In this paper we start with Maxwell's equations and then present a complete 
exposition of the theory of propagation in an inhomogeneous atmosphere where 
the dielectric constant is stratified in a radial direction. The theory can be 
applied to both ionospheric and tropospheric problems. The methods we have 
used and the results we have obtained are similar to those contained in the 
papers by Watson [1], van der Pol and Bremmer [2], and in the books by Bremmer 
[11] and Rydbeck [3]. The novelties of our treatment and results are the 
following: 

(1) The dielectric constant is considered to be a single function of position 
instead of different functions in different regions of space. As a result, the 
formulas for the Hertz potential and the electro-magnetic field have a simpler 
appearance and are not as complicated and clumsy as in the usual treatment. 
All the necessary transformations and approximations can be carried out on 
these formulas and because of their simpler appearance it is much easier to 
acquire an insight into their meaning. Of course, when quantitative results 
are wanted, it is necessary to consider the dielectric constant as composed of 
different functions in different regions, to solve the appropriate dift'erential 
equation in each region and then to match the solutions across the boundary to 
obtain the complete solution. 

(2) The expression of the Hertz potential as a contour integral and the 
transformation of the contoiu* integral into a sum of residues are carefully dis- 
cussed. There are two questions in this procedure which are ustially ignored. 
One is the question of whether the contour integral over an infinite semi-circle 
may be neglected. Watson [1] has shown that in the case of a constant at- 
mosphere these integrals vanish. However, his proof depends upon the use of 
asymptotic formulas for the Bessel functions. We show in the Appendix that, 
even for a completely arbitrary atmosphere, these integrals vanish. The second 
question is this: In general, the contour integral can be expressed as a svmi of 
residues plus an integral aroimd the imaginary axis; when can this integral be 
neglected? Watson [1] shows that, in the case of the constant atmosphere, 
this integral is negligible compared to the residue series. Later ^^Titers on 
this topic either drop the integral without comment, or change the boimdary 
condition at the center of the earth [11], or express the Hertz potential in terms 
of incident and reflected waves [2], which result only in a residue series and not 
in an integral. This latter procedure can be applied to the general case, but it 
is only in the case of the constant atmosphere that the integral over the imaginar}- 
axis does not appear. In the appendix we consider an arbitrary atmosphere 
and show, first, that if the earth is assumed to be a perfect conductor, the in- 
tegral over the imaginary axis vanishes identicall}^; and, second, that if the earth 



J 



BERNARD FRIEDMAN (S3 19) 319 

is not assumed to be a perfect conductor but if its dielectric constant is assumed 
to be large compared to that of the atmosphere, which is usually the case in 
practice, then the integral over the imaginary axis is small compared to the 
first term of the residue series. 

(3) The general nature of our study emphasizes the fact that the propaga- 
tion problem reduces to the solution of the eigenvalue problem for an ordinary 
differential equation. This fact, while it is clearly recognized in the flat earth 
theory and while it is implicitly contained in most previous work, has heretofore 
not been explicitly stated for a spherical earth. Once this fact is known, it is 
clear that the W.K.B. method should be applied. By means of this method, 
the "phase integral" method of Eckersley [10] can be understood and made 
more exact. 

(4) It is shown that the flat earth theory follows from the exact spherical 
earth theory by making suitable approximations in the ordinary differential 
equation mentioned in (3). From the nature of the approximations, it is con- 
cluded that the eigenvalues will be closely approximated but that the expressions 
for the field on the flat earth theory will be inaccurate at large distances above 
the earth. These conclusions agree with those of Pekeris [7] in his analysis of 
the accuracy of the flat earth approximation. 

(5) In the case of tropospheric propagation the W.K.B. method gives 
explicit results. First, if the atmosphere is inhomogeneous but does not have 
a duct, the field can be obtained by a change of scale from that produced by a 
constant atmosphere. This result generalizes a similar result obtained by 
Eckersley [10] and Bremmer [11] for a special form of variation of the dielectric 
constant. Second, if the atmosphere does have a duct, the differential equation 
that must be considered will have two transition points and its solution will de- 
pend on confluent hypergeometric functions. We find that the differential equa- 
tion can be approximated by one which has a double transition point and that 
then the solution can be expressed in terms of Hankel functions of order one- 
fourth. This approach seems to be completely new and suggests a simple, direct 
quantitative method for approach to the problem of propagation in a duct. 

The problem of propagation in an inhomogeneous atmosphere is formulated 
mathematically in section 2 and it is shown that the problem requires the 
solution of a scalar partial differential equation. In Section 3 this equation is 
solved by the classical method of separation of variables and expansion in terms 
of Legendre polynomials. In Section 4 this solution is represented as a contour 
integral and evaluated by the calculus of residues. These residues are inter- 
preted in Section 5 as an expansion of the solution in terms of radial eigenf unc- 
tions. In Section 6 we use the fact that the wavelength of the radiation is very 
small compared to the radius of the earth to obtain simpler boundary conditions 
at the earth's surface and a simpler expression for the solution. We return in 
Section 7 to the discussion of the radial eigenfunction and we show how, b}^ the 
use of Langer's extension of the W.K.B. method, useful approximations to the 
eigenf unctions can be found. 



320 (S320) ELECTROMAGNETIC WAVES 

The case of a uniform atmosphere is discussed by this method in Section 8 
and the well known results are obtained. Section 9 shows that the case of the 
non-uniform atmosphere without ducts can be treated almost as easily as the 
case of a uniform atmosphere. In Section 10 the "fiat earth" theory and the 
modified index of refraction are shown to be reasonable approximations to the 
exact theory. Section 11 discusses the case of an atmosphere "v\4th ducts; Section 
12 summarizes the work for the electric dipole. The detailed mathematical 
questions concerning the contour integral representation are discussed in the 
appendices. 

2. Formulation of the Problem 

Assume the earth is a sphere, radius a, with complex dielectric constant ei . 
Introduce spherical polar coordinates, r, 6, (f>, where r is the distance from the 
center of the earth, 6 is the co-latitude and the longitude. We assume that 
the earth is immersed in a medium whose dielectric constant, e, is a function of 
r alone. The propagation problem depends on finding the field produced by a 




Figure 1 

unit dipole which is radiating at a constant angular frequency w and is located 
at a distance, 5, from the center of the earth. By a suitable rotation of the 
coordinate system we can always locate the dipole on the polar axis as indicated 
in the above diagram. 

The electromagnetic field produced by the dipole should satisfy ^Maxwell's 
equations, both inside and outside the earth, and should possess at the point D 
a singularity corresponding to that of a dipole while on the earth's surface the 
tangential components of the electric field E and the magnetic field H must 
be continuous. 

Let E exp {—io)t} represent the vector electric field intensity and 
H exp I —ioit] the vector magnetic field intensity produced by the dipole, then 
Maxwell's equations in M.K.S. units become the following: 

V X E = ^coMH, V-H = 

(1) 

V X H = -zcoeE, V-€E = 0. 



BERNARD FRIEDMAN (S321) 321 

Here, e = e(r), is a scalar function which represents the variation of the dielectric 
constant of the medium. For r < a, e(r) is assumed to be a constant, €i repre- 
senting the complex dielectric constant of the earth. 

We shall derive the well known fact [12] that in spherical coordinates the 
electromagnetic field given by (1) can be obtained from a Hertz vector which 
has only its radial component non-zero, even when the dielectric constant is a 
function of r. Since eE is a vector whose divergence is zero, we may write 

(2) eE = iojn V X cC. 
Substituting this expression for E in the curl H equation, we get 

VX(H - cuVeC) = 

so that the expression in parenthesis must be the gradient of some scalar, which 
we call xp. We have then 

(3) H = coVeC -F Vxp. 

Using (2) and (3) in the remaining equations of (1), we obtain the following 
equations for C and rp: 

(4) V X ^ (V X eC) - coVeC - V^ = 

(5) V-(coVC) 4- W = 0. 

Note that if we take the divergence of equation (4), we obtain equation (5) so 
that it is sufficient to satisfy equation (4). 

Let us express these equations in spherical coordinates. Since e is a func- 
tion of r only and since the dipole is on the polar axis, the electromagnetic field 
and therefore also the Hertz vector C will not depend on the longitude </> but 
only on the coordinates r, 6. We assume that the vector C has a radial com- 
ponent of the form rU{r, 6) where C/(r, d) is some scalar function, and that its 
other components are zero. The ^-component of (4) impHes that 

so that we may take \p = d(rU)/dr and then the r-component of (4) reduces 
exactly to the following: 

(6) AU + k'U = 
where 

(7) e = co'eM 
and A is the Laplacian in spherical coordinates. 



i 



322 (S322) ELECTROMAGNETIC WAVES 

The field components can be obtained from equations (2) and (3). We 
find that 



2 , d'\, .,, „ 1 d' 



(8) 



H. = [^-^^)(rUh H^ = ftee^rU), 



E, = -ic^f^j^, Er =E,=H^ = 0. 



The equations in (8) represent the field of a magnetic dipole. There are similar 
expressions for the field due to an electric dipole. The modifications of the 
theory for this case are detailed in Section 12. 

The problem of the propagation of radio waves around the earth now has 
been reduced to that of solving the scalar wave equation (6). We know that 

/:'(?') = constant = Je^ii, r < a 

lr(r) = ue(r)fi, r > a 

and that, at r = a. U and dU dr must be continuous because the tangential 
components of the field given by iS) must be continuous. We know also that 
U must be regular ever^-^'here except at the dipole. that is. at r = 6. ^ = 0. 
Finally, we know that the dipole is sending out radiation so that all the energy- 
is going to infijiity. This is expressed mathematically b\' the Sommerfeld 
radiation condition. Before ^Titing it do^^Ti. we remark that since the medium 
must eventually become free space, the value of e ^ill approach that of free 
space which we designate by e, . The Sommerfeld radiation condition may now 
be expressed as 

(9) limr ^ - ikoU = 

where 

(10) k'o = co'coM- 

This condition means that, at infinity. U behaA'es like exp {ikQr]/r so that we 
are dealing with an outgoing wave. 

3. The Series Solution for L(r, 0) 

The fact that U[i\ 6) possesses a dipole singularity at r = 6, ^ = while 
otherwise it satisfies the wave equation (6) can be expressed very concisely b}' 
using the Dirac d-function. We write 



I, 
BERNARD FRIEDMAN (S323) 323 \ 

where the denominator 27rr^ sin B has been introduced to normalize the solution. 
It is essentially the Jacobian of the transformation from rectangular to spherical 
coordinates. 

Since U{r, 6) is a regular function of {r, 6) except at r = 6, = 0, we may 
assume 

(12) U = T, ^^^J^ i^n(r)P„(cos 6). 

n = ^ 

From (12) by using the well known orthogonality properties of the Legendre 
polynomials we find that 

(13) w„(r) = [ [/(r, ^)P„(cos Q) sin UQ, 

•JO 

Now, multiply (11) by P„ (cos 6) sin Odd and integrate between and tt. 
The equation transforms into the following ordinary differential equation; 



> (14) 



r dr L r A i 



b) 



or 



^^ + [m - ^^^](r.J = - 



8{r - h) 
27rr 



Let us investigate the meaning of (14). If we integrate it first from r = Tq 
to r = h — h and then from r = Tq to r = h -\- h, h a small quantity, we find 
that d{rUr)/dr has a jump of magnitude — 1/2x6 as r goes from h — h to h -\- h, 
while rUr, remains continuous. Note that the point r = a causes no trouble 
because, there, just as for every value of r ^ 6, the solution ru^, and its derivative 
must remain continuous. (These conditions are, of course, the same as those 
which have already been imposed on U{r, 6).) The other conditions on U, 
that it be regular at r = and that it satisfy the Sommerfeld radiation condition 
(9), imply that Un is regular at r = and that 



(15) lim 



, ZfCoTUn 



0. 



In order to solve (14) we must know the solutions of the homogeneous 
equation 

(16) m + [,.(,) _ ?^(--^i)](„) = 0. 

We shall study this equation, not only for integral values of n which are the 
only ones needed to solve (14), but also for complex values of n which will be 
needed later. In case k^(r) is a constant, kl say, the solutions of (16) are arbitrary 



324 (S324) ELECTROMAGNETIC WAVES 

linear combinations of r^^V„+i(fcor) and r^^^Hnll(kor). The function r^^^J„+j(A:or) 
is regular at the origin if real part of n > —1, while the function r^^^H^l[{h;r) 
satisfies the radiation condition. If k^ is not a constant, the solutions of (16) 
will behave approximately like the Bessel functions but their exact beha\'ior 
will depend on the way k^ varies with r. 

We shall introduce three particular solutions of (16). Let rVr,{r) be that 
solution which is regular at r = 0, while rwi^\r) is that solution which for large 
r is asymptotically exp {ikof}, and rwl^\r) is that solution which for large r 
is asymptotically exp {—ikor}. Note that the function rwl^\r) satisfies the 
radiation condition (15). For simplicity of writing, we shall write it as rWr^r) 
when there is no possibility of confusion. 

Since the differential equation (16) is unchanged when n is replaced by 
— n — 1, we see that 

This relation is not true for v^ir) because the condition of regularity at the 
origin will imply that f„(r) behaves like r", n > 0, at the origin and this con- 
dition is changed when n is replaced by — n — 1. It should also be noticed 
that in the case where k^ has a constant value kl throughout, the function 
rVr,(r) may be taken as 

r''V„.^(M; 

on the other hand, because of the behavior at infinity, we must renormalize 
rwl^\r) as follows: 

rwi'\r) = (l)''' exp 1+^4) exp l+i{2n + l)T/2Kkory''Hi%,(kor) 
and 

rw':\r) = (l)''' exp {-^V/4} exp {-^(2n + l)ir/2]{kory''H:iU{kor). 

We shall now prove the well-known fact that the Wronskian of any two 
solutions of (16) is a constant independent of r. Consider, for example, the 
two solutions rv^^ir) and rw^ir). We have 



jK)Hh[^^-^:ti)](n;„) =0 
(rw.) 4- ^k' - ^^^^^^^](ri/^n) = 0. 






Multiply the first equation by rw?„ , the second by rVr, and subtract. W^e get 

(rw^) —2 {rv„) - (rvr) ^ (m„) = 0. 



BERNARD FRIEDMAN (S325) 325 

Integrate this equation from r = r^ to r = r^ , and we have 
t \rwn{r) j^ (rvn) - rVr,{r) — (r^J J = ^rw^ir) ^ (Wn) - rVr,(r) — (rw;„) J 

so that the Wronskian 

t (17) Wn = rWn(r) — (rVn) — rVn{r) j- {rWr) 

is independent of the value of r but depends only on the value of n. 

Now that we have some knowledge of the solutions of (16), we can prove 
that the solution of (14) is given by the following formulas: 



(18) 

^ rv^(h)Wn(r) 
2TrW^ 



r > h. 



First, it is clear that both expressions in (18) satisfy (14) for r ?^ h. Second, 
at r = 6, the function ru,, is continuous and its derivative has a jump of magni- 
tude — l/27r6. Third, from the definition of the functions Vn{r) and Wn(r), the 
function w„ is regular at r = and satisfies the radiation condition (15). This 
shows that rUn(r) really is a solution of (14) satisfying all the other conditions. 
Finally, we can obtain a representation for U{r, 6) from equation (12). 
We have 

U(r, e) = l± "-^ ^ P,(cos e), r<b 

(19) 

v„(b)«)„(r) 2w + I 






This is the solution of the problem in its most general form for an arbitrary 
variation of the dielectric constant. Special examples of this are well known 
for the cases in which the ordinary differential equation (16) can be solved in 
terms of known functions; for example, Rydbeck [3] has treated the problem 
where e(r) is assumed to vary parabolically. It is to be noted that this solution 
(19) can be used for computation only if ka is small compared to unity. In 
the following sections this solution will be transformed so that it can be readily 
computed for large values of ka. 

It is interesting to consider some special cases of (19). Suppose, first, that 
the medium is uniform and that there is no earth so that k^ = kl , a constant, 
then using the standard notation for the spherical Bessel function [13] 

Vn{r) = Uhr), w^{r) = r'h':\k,r), TF„ = iVK 



326 (S326) ELECTROMAGNETIC WAVES 

formula (19) becomes 



(20) 



U(r, e) = ^'^P^jl"^' = + f^' E (2» + l)i„(fc„r)fe"'(fco6)P„(cos 9), r < 6 
= + T^ Z (2n + l)Ukob)K"{kor)P,{cos 0), r > b, 

4x 



the classical formula for the dipole [13], where R is the distance from the point 
with coordinates (r, 6) to the source point (b, 0). Suppose now that the earth 
is present so that for r < a, k^ = kl , r > a, k^ = kl ; then 

Vnir) = jnikiv), r < a 

Vnir) = ynjuihr) + djin^kov), r > a 

w^(t) = i-'-'C\kor), r > a 

where 7„ , dn are to be determined by the fact that w„(r) and (d/dr) Ur,(r) are 
continuous at r = a. From the definition (17) of the Wronskian, it follows that 

TT. = C''h'yMK'\kohmkoh) - K'"{k,h)j.{k,h)] 

= ilJk, . 
7 is found from the following equations: 

dn{K0) = ynjnikoO) + d^'^koO) 

hf„{k,a) = ynkofnikod) + 8,koK'''(koa). 
We have 



h f^{k,a)K'\k,a) - koUk,a)h'^'\koa) 

7„ = 

so that 



^" K'\hamk,a) - hl!'\k,a)jAa) 



(21) Wn = r'-Xa'lk,fr,{k,a)K'\kod) - koUk^a)K'\koa)]. 

Finally, 






r < a 



= f E (2» + 1) T''J"(^»'-) +/-^"'fe'-) r-/,^"(A-o6)P„(cos 6), 
47r ** n 

(22) a <r < b 



BERNARD FRIEDMAN (S327) 327 



47r ^ n 

r > h. 

Notice that jn/Wn = ^o^ " so that, by using (20), (22) can be written as the 
sum of two terms: 



, ^ exp {ikoR] 

^' ^ 47rR 



(23) 



+ ^ Z (2n + 1) M^iM rX'\kor)P^icos e), r < h. 



The first term is due to the dipole while the second term is due to the scattering 
of the dipole field by the earth. 

In case the earth is assumed to be a perfect conductor so that ki =0°, 
jnikid) = and, from the continuity condition, 5„ = —ynjn{koa)/hn^\kQa). We 
find then that 

Uir, 0) = g E (2r^ + l)[Ukor)hl!\koa) - hl'\kor)Ukod)] 



(24) 



C\koh) 
h':\k,a) 



^^^,, ^P^icosO), a <r <h 



5 Z (2n + l)[jnikohK\koa) - hl'\kob)jAa)] 



h'n\kod) 



^^ ^ P.(cos e), r > b. 



4. Contour Integral Representation for U(r, 6) 

Following a method due to Watson [1], we shall show that the series (19) 
can be expressed as a contour integral. Consider the series 

(25) S=Y, ?^^ a.P„(cos 6) 

^ 

where a„ is assumed to be an analytic function of n. We shall show that S 
can be represented by the following integral: 



(26) I = — \ (^+ |)a.P.( - cos 6>) 

27rz Jc sin irt 



dt 



328 (S328) 



ELECTROMAGNETIC WAVES 



where C = Ci + C2 is a contour that starts at 00 — ih in the ^-plane, goes below 
the real axis to ^ = — 1/2 and then above the real axis to <» -]- i^. 



^- 


t- plane 
<^2 


"-V 


' 




/ 2 3 


c, 



Figure 2 



Since a< and Pt ( — cos B) are analytic functions of t, the only singularitie, 
of the integrand are poles at those values of t inside C for which sin tt^ = Os 
that is, i = 0, 1, 2, • • • . Since the contour C is described in the clockwise 
direction, the integral 7 is equal to the negative sum of the residues at the poles, 
so that 



(27) / = - Z 



(n + i)a„P„(- cos 



TT cos -KU 



- = - ; i: (n + |)a.P«(cos B) 



because P„( — cos Q) = ( — 1)"P„ (cos 6). 

Formula (27) shows that S = —tI. If this procedure is applied to (19), 
we find that 



(28) 



U(r, e) = 



47rz Jr. 



(^ + \)v,{h)w,{r)P,{-co^e) 



W. 



sm irt 



dt 



where C is the contour specified above. On Ci , the part of C below the real 
axis, replace / by — / — 1. then Ci is transformed into C3 and we have 



(29) 



_J_f 



{t + i)v-t-^{h)w.,^,(r) P„_,(-cosg) 



W 



sm tt/ 



cff 



(^ + i)Vt{h)wXr) P,(-cos 0) 



1^. 



sm irt 



dt. 



We shall rotate C2 and C3 so that they go around the upper half of the 
line, imaginary part of t = —1/2. To justify this, it is necessary- to show that 
the integrals^ over the infinite quarter-circles in the first and second quadrants 
go to zero. This is discussed in the appendix. 

When the contour is rotated in the desired manner, it ma}' pass across 
some poles of the integrand. We find then that U{r, 6) as given bj' (29) can 
be written as the sum of residues at the poles plus two integrals over the upper 
half of the line ^m (t) = -1/2. Put t = -1/2 + it and use the fact [14] that 

(30) P,(-cos e) = P_,_i(-cos e), 



BERNARD FRIEDMAN 



(S329) 329 



we obtain 

(31) 



e)=y residues + -^ f ^P^r-,{-cos 6) p_^(b)^..._,(r) 
, (f) Z. residues + ^^. j^ _ ^^^^ ^^ ^ ^^^ ^ 



V.ir-^{b)W 






W. 
From (28) it is clear that the poles of the integrand will be at the points 



ImffJ 




Figure 3 



where Wt = 0. Call these points U , tz , 

is 



then the residue at any pole, t, say, 



(32) 
Here 

(33) 



W'ti sin irtj 



wu = j^ w. 



We thus have the final result: 

TUr A\ - V (<- + i)v,,(b)w,,(f)P,X~cos e) 
^^' "> - '^ TF;, sin Tt, 






tP,>-^(— COS 



^iri Jo — cosh TTT L ^ir-i 



Li)[.. 



.i{h)Wir-h{r) 






5. The Eigenvalue Pro'blem 

The residue in (32) can be more usefully interpreted in another way. 
Since the Wronskian, W^ = 0, the two functions Vt^(r) and w,-{r) are not 
independent, that is, there exist constants aj such that Vt^ir) = ajW,.(r). This 
equation shows that the function Wt^ir), besides satisfying the radiation condi- 
tion at infinity, also satisfies the condition of regularity at r = 0. In other 



L 



330 (S330) 



ELECTROMAGXETIC WAVES 



words, tj is an eigenvalue and Wt,{r) is an eigenf unction of the follo^sing problem: 
Find the values of t for which v(r), the solution of the following differential 
equation: 



(34) 



d\rv) 
dr' 



+ [;/ - ^(^](n.) = 0, 



is regular at ?' = and satisfies the radiation condition (15) at infinity, that is, 

,. d(rv) 

lun I -V^ — ikoTv ' = 0. 

.^co 1 dr 

Put 

(35) ^,- = f w.XrYdr 

so that iVti(r)/^Y^ is a normalized eigenfunction of (34); then the residue given 
by (32) will be shoT\Ti to be equal to 



(36) 



w,,{h)w,,{r)P,X-cos 6) 
2/3y sin -Kti 



This result will be obtained from a consideration of the following two equations: 



dr 



+ 



d-(rWf,) 
dr 



+ [,= _ Mk^](,,,,.) = 0. 



Here / has an}^ value and is not an eigenvalue. ^Multiply the first equation by 
rWtj , the second by rvt subtract and integrate with respect to r from to ^o . 
We obtain 

(37) {t - t,){t + ti + 1) I i\Wt. dr = Wt -T, {rw,;) - nv,. j {n\) 



Notice the term at zero drops out because b}' definition r, and u\, are regular 
at /' = 0. 

Consider the followino; determinant: 



n\ 



n\ 



rwt 



rWi 



rWt 



rw. 



J,(n:) ^Aru,,) j^(™„.) 



BERNARD FRIEDMAN (S331) 331 

It is obviously zero since the elements of the first two rows are identical. When 
it is expanded in terms of the elements of the first two rows, we have 

rvt\ rwt -r (rWt,) — rwt^ -r (rWt) 4- rwt\ rvt j- (rwt) — rWt -r (rVt) 

— rwt\ rwt^ -r (rvi) — rVt -r (rWt^) =0. 

Now, as r approaches infinity, the first bracket approaches zero since rWt and 
rWt , both satisfy the same condition at infinity. This shows that 

p 1 w, [rw.f^(rv.)-rv.f^{rw.)\ 

SO that (37) can be written as follows: 

As t approaches t, , Wt approaches Wt^ and Vt approaches ajiOt^ while the right 
side of the above equation becomes indeterminate. If the right side is evaluated 
by L'Hopital's rule, we find that 



«/ / '^tiiry dr 

Jo 



W[ 



2/y + 1 

SO that 



^i = 



(2^, + l)a, 



using this in (32) we get (36). 

When (36) is substituted in (31) we get finally the result that 

(38) w, e)=j: «?i^(|^ f^ 4^0^ + J 

Zfjj sm TTij 

where 7|is the following integral: 

7 = JL f " tP,>_^(- cos e) r v.ir-h(b)w^ir^h(r) _ Vir-h{h)Wir-ii{r) } ^^ 
47^^ Jq cosh ttt L W-ir-\ Wir-^ J 

In the appendix it will be shown that, if the earth is assumed to be a perfect 
conductor, then / vanishes identically. If the earth is not assumed to be a 
perfect conductor, / will not be zero. However, it will be shown that I is small 
compared to the first term of the series in (38) as long as the dielectric constant 
of the earth is large compared to the dielectric constant of the atmosphere. 



332 (S332) 



ELECTROMAGNETIC WAVES 



6. A Simpler Form for the Hertz Potential 

So far, we have not yet made use of the fact that the dielectric constant of 
the earth is a constant, nor of the fact that, for the wavelengths we are interested 
in, ka is very large of the order of magnitude of 10^. We shall use these facts 
to simplify the expansion given in (38). 

It will be seen in the later sections that the zeros of the Wronskian, t,- , will 
all be very large, of the order of magnitude of ka. We are therefore justified 
in using an asymptotic formula for the Legendre functions that appear in (38). 
We have [14] 

^'(<=- *) ~ U + lsineT -^ {{' + I)* - i} 



for large values of v, so that 



(40) 



P,(— cos d) _ Pt,{GOS {it - 

sin irtj sin tIj 



e)) 



L{t< +% sin e) ' '''^P v('' + i) 



Next, since for r < a, k^ = kl = constant, we know that the solutions of 
(34) are 



rv 



1/2 



r^'^Hl'Mhr), 



r < a. 



r-Jt+^ikiv) or r rLt+x\ 

The function Vtir) was defined to be that solution of (34) which was regular at 
r = 0. It must, therefore, be proportional to the function J t+i^ikir) for r < a. 
Since the solution of (34) and its first derivative must be continuous at r = a^ 
this requires that 



(41) 



Put 



(d/dr)rvt{r) 
rvt{r) 



id/diWJ,,.{k,r)) 



r'''J..^{k,r) 
Again, the right side can be simplified by using the asymptotic formulas. 






SO that 



r'''J.,x(k,r) 



C exp I j g{r) 



dr 



where C is a suitable constant. Substituting this expression in the differential 
equation for r^^^Jt+^{kir), that is, 



d'jrv) 
dr' 



+ [^? - ^^](-) = 



BERNARD FRIEDMAN 



(S333) 333 



we get 
<42) 



^ 4_ ^2 , . 2 __ t(t + 1) _ ^ 



Since /cj = co^eiju, where ei , the dielectric constant of the earth, is very large, 
we may neglect dg/dr in (42) and get the approximate formula: 



.(.) = ±{.? - ^^]' 



To determine whether it is a plus or minus sign, we notice that ei and so, also 
^1 , has a large positive imaginary part so that, by the usual asymptotic formula, 



r^'V..j(^,r) - (?)''' cos|A;,r - | (^ + |) 



-> r^ C2 exp l-ik^r] 



where C2 is independent of r. Then 



(d/dr)(r'''J,,^(k,r)) 



iki 



and so g{r), which is a more accurate expression for the ratio of the derivative 
to the function must have the minus sign, that is, 

g{r) = -i[kl - t{t + l)/rY\ 

We shall see later that t ^^ ak(d). Because of this, g(a) = —i{k1 — k^Y^^, 
a quantity independent of L Now (41) can be written as follows: 



(43) 



{d/dr)rvt(r) 
rvt{r) 



= -i{k\ - k'Y' 



This result holds for all values of t, if t is approximately ak(a). 

Consider the eigenvalues t,- for which the functions Vt{r) and Wt{r) coincide. 
Then (43) gives us the following eigenvalue equation: 



(44) 



{d/dr)rwti{r) 



m - ky'\ 



We shall discuss its solution in a later section. 

There is one more quantity in the expansion (38) that will be simplified. 
It is the quantity jS, or essentially W' . Since the Wronskian is constant for 
all values of r, we have 



TT. 



d f . 
rwt -r irvt) 
ar 



rvt ^ (rw,) 



J TTT, dirwt) d , . , d d . . d(rvi) d , . d d . . 



334 (S334) 



ELECTROMAGNETIC WAVES 



all evaluated at r = a. Now, since the right side of (43) is independent of t, 
we have 



d id/dr){rvt) 
dt rvt 







or 



d d . . 



d(rvt) d 



{rvd 



= 0. 



dt dr 
When t = tj , an eigenvalue, the function y«,(r) is identical with iP(,-(r) and then 

Using this result in the formula for W[ , we find that 

d d 






-™.,-^(™,.) 



If Wti{a) ^ 0, which by (44) will always be the case if \:^ 9^ co , that is, the earth 
is not a perfect conductor, then w^e may write the above formula as follows: 



(45) 



W[. 



(^^^ y A {dldr){rw,:) 



dt 



rw. 



= {aw.fM, 



W'here M is defined by the last equation. 

Combining the results we have obtained in (45), (44), (42) and (38), we 
get, finally, the desired expression: 



U{r, e) = 



y L TT sm ^ J ' 



('.-i) 



(46) 



tf;, 



6r y L TT sin ^ J M aWt^o) awt^(d) 



Here, hw t ^{h) / aw t ^{0) is the height gain factor for the transmitter and 
rwtj{r)/awtjia) is the height gain factor for the receiver. 



7. The Longer Asymptotic Formulas 

The previous sections have sho\\Ti that the problem of determining the 
Hertz vector U{r, 6) and thus the electromagnetic field depends upon the solu- 
tion of the eigenvalue problem for the differential equation (34). In general, 
this problem has to be solved by some approximation. In this section we shall 
discuss by a method due to Langer [9] the well-known W.K.B. approximation 
to the solutions of a second-order differential equation and in later sections we 
shall use these approximations to obtain the eigenvalues of (34): 



BERNARD FRIEDMAN (S335) 335 

(47) ^ + [k(rr - f\(rv) = 0. 

Here, we have put X for t(t + 1). 

Consider a second order differential equation of the form: 

(48) gi + [^VW + 9ir)]s = 

where v is assumed to be a large constant and where f(r) = (r — ro)°'fo{r), a 
is a constant and/o(ro) ^ 0. Put 

(49) r- ro = h{r), s = aQi^r))'^' 
where /i(r) is so chosen that 

(50) h'irffir) = 1 

and r = for r = To . Under the transformations (49) equation (48) becomes 

where \1/{t) is small for large values of v. The solutions of (51) are, approxi- 
mately, linear combinations of 

(52) t'''H\'\vt) and r'''H\'\vr) 

where y = {a -\- 2)~\ 

We shall see later that for our purposes only the cases a = 1 and a = 2 
are of interest. The case a — \ corresponds to the situation where the atmos- 
phere does not have a duct or a temperature inversion. The case a = 2 will 
correspond to the situation where the atmosphere has a duct. 

In case a = 1, we rewrite equation (47) as follows: 



d\rv) 






4(S)" - sl>' = »• 

This equation is of the same form as (48) if we make k^a correspond to v and put 

^('■^ = ll^j -Jm^' »W = 0. 

To use the formulas (52) we must put from (50) 

dr 1 



dr h\r) 



imr or 



^^^^ ' = llA\h) "(^ 



1/2 

dr. 



336 (S336) 



ELECTROMAGNETIC WAVES 



Here, ?'o is taken to be a zero of /(r) so that ro/c(ro)^ = X. If we change also the 
dependent variable from v{r) to (t{t) by the transformation indicated in (49): 



\koJ (korfj ' 



rv = (tW{t)Y'' = 

then, according to (52), we have 

^^,,(1.2) 1/2 f k{r) \ X/a r7(i,2)/7, ^ N 

In particular, the solution which satisfies the Sommerfeld radiation condition 
at infinity will be given by the formula: 



(54) 



rw. 






It should be noticed that the W.K.B. method and the Langer formulas are 
not valid if in (48) /(r) is discontinuous. This means that formula (54) is not 
valid for r < a. However, we have already seen that the solution of (34) for 
r < a behaves like the Bessel function Jt+\(kxr). Using the approximate formula 
(54) and neglecting terms of order {ka)~^, we obtain a simpler form for the eigen- 
value equation (44). It becomes 

^^^> l\kj (hafj i/,<»(fcoarJ ^oL \kj 

Here t„ is the value of r corresponding to r = a. Note that, if the earth is 
assumed to be a perfect conductor so that A;, = <» , then equation (55) reduces 
to the following simple form: 



(56) 

Since the ratio 



HDlikoar:) = 0. 



dH\)l{x) 
dx 



/h\) 



lix) 



will appear so frequently in our analysis, we shall use for it the simpler notation 
Z{x). Equation (55) can now be written as follows: 

[©■-i^r-<'--'=^['-(f)7" 



Put 



y = C,H[)l{x) = exp I J Z{t) 



dt 



where Ci and a are suitably chosen constants. From the differential equation 
satisfied by y 



J 



II! 



2.1^.- 1 



Z' + Z^ + - Z + 1 - ^2 
X 9x 



BERNARD FRIEDMAN (S337) 337 

we find that Z satisfies the Riccati equation: 

so that for large x 

(57) Z' ^ l-Z'-iz. 

X 

8. The Case of a Uniform Atmosphere 

We shall discuss first the case of a uniform atmosphere in order to more 
easily understand the case of a non-uniform atmosphere. The methods and 
results of this section are classical [11]. Suppose that the atmosphere is uniform 
so that k^ = kl = constant and also that the earth is a perfect conductor. The 
solutions of 

Hl%iO = 
are 

(58) ^0 = 2.38 e"'^', ^^ = 5.6 e""', ^2 = 8.65 e"'^\ etc. 
The equation (56) now becomes the following: 

^''^ ^° £ b - v^T * = «" = ^""^" ' 

where ^„ is defined by the last equation. 
Put 

(60) p = kor\-''\ p„ = koa\-''' 
in (59). Since koVo = X^^^, it reduces to 

\''' p (1 - p-y dp = e-'^V^ . 

When we evaluate the integral we get, finally, the eigenvalue equation, 

(61) (pi - ly' - arctan (pi - 1)''' = e-^'rj.-'''. 

In this equation Pa is a function of X defined by (60). 

We shall find that X^^^ is of the order of magnitude of (koo) and therefore 
the right side of (60) will be very near to zero which implies that pa will be close 
to one. 

Put p„ = 1 + dikoa)-^^^ then (61) becomes (25)'^'(/coa)'V3 = e""V„X~''' = 
e"''V„(A:oa)"^ or 

(62) 5 = K3T„)'/'e-'''■/^ 

Each value of t„ determines a value of 8, then a value of p^ , and by (60) a value 
of X. Since t is approximately X^^^, we finally get the well-known expression [11] 
for the eigenvalues: 



L 



338 (S338) 
(63) 



ELECTROMAGNETIC WAVES 



tr. = koa + i{hay'\3r^y'V'' 



Note that (62) shows that 6 is of the order of magnitude one and since 
{koay^^^ is small, our assumption that Pa is close to one is justified. It is also 
important to notice that the integral below (60) can be easily evaluated ap- 
proximately. Since pa is very close to one, we may proceed as follows: 

j'' (1 - p-Y' dp ~ j'' (p^ - l)-'^ dp ~ 2*- l" (p - 1)"^ dp 



= I 2''\p^ - If' = \ {2Sf\har 



which is the same result that was obtained by an exact integration. 

If we drop the assumption that the earth is a perfect conductor, we must 
solve the eigenvalue equation (55) instead of the equation H[)\{^) = 0. Using 
the same approximations and notations as before, we find that 

Ka-r = fc„ £ [l - ^^] dr 

pPa \l/2 1 



and that 



~ ■— (koar\2sr' 



(2S)' 



so that (55) becomes 

-l/3/o^\l/2yj (26)^ 



(64) 



(M (25)^^'Z' 



I 3 






J [■-(¥)■; 



= 77' 



where rj is defined by the last equation. Methods for solving this equation 
are discussed by Bremmer [11]. 

We shall now find a simple approximate formula for the quantity M which 
appears in (45) and then we can write down the complete formula for the Hertz 
potential as given by (46). From (45), (57) and (64) we find that 



M^ 



d^ (d/dr)(rwt,) 
dt rWti 



= /co(M ' t: 



-ib'y'AT}] 



-'•<M-'l'f,[<-)"-4T}] 



= ko{kod) ^' 



dd 
dt 



db 



- koihay''' ^ 1(^28)''' ^ 



L(25) 
L(26) 



175 Z + 25Z'J 

25<1 + Z' + 



(26) 



3/2 



-^[25 + 25Z^ + ^2] 



dt 



BERNARD FRIEDMAN (S339) 339 

= —kQikoO) — 25 + r; (/Coa) + ^^ 1. 

Now, by (60) 

^(^ + 1) = X = (M'p:' = (M'[i + 5(M"''1"' 

so that, approximately, 

t = koa[l + 8(koay~^T' = k^a - Sfea)'/^ 
and then 



e''/^ 



^n = y^ e^P ix(3r,)^^^(e--0/4} 
As was pointed out before, these results are well known . 

9. The Case of a Non-uniform Atmosphere 

If the atmosphere is not uniform and is sueh that r^k{rY has no stationary 
point, that is, physically the atmosphere does not have a duct, then it can be 
treated similarly to the case of a uniform, atmosphere. We shall see that the 
results of the preceding sections will be only slightly modified. 

First, we find an approximate formula for r as defined in (53). Since a will 



11= -(M- I 

Using this result in the expression for M given above, we find, finally, that ; 

(65) M = ko{koay'[28 + r,'(koay'' + rjikoaY^'d-'], Ij 

t! 

Consider the case where both the transmitter and receiver are on the earth's . j 

surface so that b = a and r = a. Substituting (65) and (63) in (46) we get 

U{a, d) = - ^^P ^'Y - [2^iikoay''df'' 

TTCLU \ 

(66) 

y. exp m,aY"e{ZT,f\e'"")m ^an 

Here t, is the root of (64). Note that a6 = d is the distance from the transmitter 
to the receiver so that 

U(a, 0) = - 55Pi|»^ (2«x)''^F(x) i' 

Td 

where x = ihaY^^d 

,2/3/Tri/S 



340 (S340) 



ELECTROMAGNETIC WAVES 



be near to Tq and X^^^ to ak{a), the bracket in (53) may be approximated as 
follows. Put 



Then 



(67) 






k{a) = K , [rk(r)]U = f, ^Kko' = K\ 
t%{tY - X 2ah 



{koof (kod) 

2aka 
{kod}'' 



[rkir) - \'''] 



«^-^°)~2^(:--i)=Ht-0 



Here, we have made use of the fact that since roA:(ro) = X^^^, rk{r) — X^^' = 
r(^ — ^o) approximately. 
Now, 

koar = 2''%K f (^ - l)''' dr = 2'''Kkoa(^- - if'/S. 

If we put, as before, a = ro[l + d{Kkoay^^^] then koar, = (25)^'''/ 3 and the 
eigenvalue equation (55) becomes 

(68) {Kkoar'^26y"z\^^ = -i ;^ [l - |j]'" = v, 

where rj is defined by the last equation. 

Now t '^ X^^^ = rok(ro) = ak^ + ^{vq — a), by Taylor's Theorem, but 
Tq — a - - 8ro{Kkoa)~^^^ ^ -aSiKkoay'''^ and so 

(69) t = ak^ - b{Kk,ay'\Kko/K). 

In most cases the term ak'{a) will be negligible compared to k^ , so that 
^ = ka and K = ka/ko . Then 



(70) 



t = ak, - 5(A-,a)'/'. 



If this formula for t is compared with the corresponding formula (63) for the 
case of the uniform atmosphere, we observe that the only change is the sub- 
stitution of ka for ko . 

The value of M can be found by the method used in the preceding section. 
We have 

■''' - \{28y''7l^''^ 



M = KkoiKkod) 



dt 



I 3 



= -KkoiKkoa)-''' ^ I 25 + 25Z' + 



= -KkoiKkoO)- 



dt 

1/3 ^ 
dt 



L (28)''' ^ J 

[25 + v\Kkoar'' + ^{Kkoay''8-']. 



BERNARD FRIEDMAN (S341) 341 

Now, from (69), d8/dt = -{kJK,o){Kkoay'^^ so that j! 

(71) M = -UKkoay'''[28 + ^\Kk,af'' + ^(KM'''^"']- "^ 

Again, consider the case where the transmitter and receiver are both on 
the earth's surface so that h = r = a. Using (69) and (71) in (46), we have 

,,, ,, exp [iKae] l2 7ri{Kkoay''df' 

Uyci, u) = — ~ ; 

^«^ k^iKko)-' 



e/2 



,-2x y. exp {-i{Kkoay''eKko8/k^} , 

^ ^ * ^ 26,- + v^KkoaY'' + v(Kkoay''8-' ' 

^ _ exp {ikgd} (27rzxi)'^^ j^ ^ . 
^ad kXKko)-' '^""'^ 
where xi = (KkoaY^^d 
(yo\ F{ ) = V exp {-ixiKkod/kg} -0/2 

Note that again, if ak'(a) is negligible compared to ka , so that K = kjka , 

X. = {Kafe, F.(x,) = F(x) 
as defined in the preceding section and 

(74) . Via, 6)=- SBL^ (2xixO"^F(xO. 

This formula shows that the Hertz potential in the case of a non-uniform at- 
mosphere can be obtained from the Hertz potential in the case of a constant atmosphere 
by making the substitutions: ka for ko and 1\ for r} where ^ is obtained from 77 by 
replacing ko by ka . 

The case where the transmitter and receiver are not on the earth's surface 
can be derived from the preceding results by multiplying each term in the series 
for F(x) by the corresponding height-gain factors for the transmitter and re- 
ceiver. The height-gain factor for the receiver is 



.7,, lEn^ _ (rY'[ (akay-\ f^(rY^ H^ 

^^^^ aw,,{a) \Ta) lirkf - X J W H[% 



(koar) 



(koaTa) 

by (54). We shall introduce into this formula 6r , the angular distance of the 
receiver from the horizon, then (75) will take a simpler form. 

From the definition of dr , and since r — a is small compared to a, we have 
cos Or = a/r ^ 1 — (r — a)/a and since cos Or = 1 — dl/2, we have Or = 
(2(r - a)/ay^\ Put Xr = (akay^^Sr , then, from (67) 

iij - Wf - 2^<:- - = '^t !i + ^^"'^^r^-r] 

= K\aKr'\x\ + 2S) 



342 (S342) ELECTROMAGNETIC WAVES 

and we find that koar = (x? -f- 2 6) ^''73. Since for r = a, Xr = 0, we see that 
the right hand member of (75) becomes 



^ ^ L 26, J H[% 



+ 25)-^/731 



{(25)''73) 

There is a similar formula for the height gain factor for the transmitter. Instead 
of Xr in (76) we must use Xh where 

X5 = {aKf^'e, , e, = [2(6 - a)/ar\ 

It should be pointed out that results similar to those of this section, which 
shoAv that the field due to a non-uniform atmosphere could be obtained from 
the field in a uniform atmosphere, have been obtained by Eckersley [10] and 
Bremmer [11]. However, they had to assume a particular form of variation of 
the dielectric constant, to wit, e(r) - 1 — a + /3a^/r^ where a and /3 are small 
constants. The results of this section are independent of the form and variation 
f the dielectric constant. 



10. Connection with the ''Flat EartK^ Theory 

The preceding sections have shown that to a good order of accuracy the 
eigenvalues depend on the behavior of k{rY only in the neighborhood of the 
transition point, that is, the point at which k{rY — X/r^ = 0. It was also 
shown that at the transition point Tq = a, very closely. If these facts are taken 
into account, the differential equation (34) may be approximated as follows : 

(77) ^ + [&' - m(rv) = 

w^here N^ = r^k{ry/a^kl is the modified index of refraction and where kl = 
at^ + l)/a\ 

Equation (77) is exactly the eigenvalue equation that is obtained in the 
troposphere theory on the assumption of a flat earth [15]. If the W.K.B. method 
is used to solve this equation, we see that we get the same results as bj' applying 
the W.K.B. method to the more exact equation (34). There will be a discrepancy- 
between the two methods only at points where r differs very much from a. 
This indicates that the height-gain factors on the ''flat earth" theory will be 
wrong for large heights above the earth's surface. Another possible source of 
error is due to the fact that the Legendre functions of the exact theory have 
been approximated by exponentials. This approximation will be bad for ex- 
tremely large distances between receiver and transmitter. Similar conclusions 
on the range of validity of the ''flat earth" theory were reached by Pekeris [7]. 
On the whole, however, the results of the previous sections have shown that the 
*'flat earth" theory gives a very satisfactory approximation to the solution. 



BERNARD FRIEDMAN (S343) 343 

11. An Atmosphere Containing a Duct 

In Section 9 we have discussed propagation in a non-uniform atmosphere 
under the assumption that r'^k{rY does not have a stationary point near r = a. 
The reason for this assumption is that, otherwise, the differential equation (34) 
will have two transition points, that is, the equation k{ry — \/r^ = will 
have two roots in X near X = (aka)^. Now, the Langer approximation by means 
of Bessel functions of order one-third is valid in the neighborhood of each 
transition point separately but it cannot be used in a neighborhood of both 
transition points. Langer [17] has pointed out that in such a case the approxi- 
mation to the solutions of the differential equation should be in terms of con- 
fluent hypergeometric functions. Since these functions are not too well tabu- 
lated, we shall attempt another approach. 

In practice, the two transition points of the differential equation are close 
to each other. We shall approximate the differential equation by assuming 
that the transition points coincide so that we have a double transition point. 
In such a case we have a = 2 in equation (51) and the approximate solution 
will be expressed in terms of Bessel functions of order one-fourth. 

Let To be a zero of k(ry — X/r^ = 0. Put 

f(,) _ MlO! _ _X__ _ (r - r„) r . _ Xl' 
^^'' - kW klaV kW l^' r'L,, 

and g{r) = (r - ro)| fc(r)" - ^J ^ 

so that/(r) has a double zero at »" = ro . According to (50), 



i I-^^''^]'' 



and then from (52) since a = 2, we have that 

(78) rw, = r'''[mr'*H[]l{koar) 

The eigenvalue equation now becomes 



^-f/^Nl/2 ^l/4:{koCiTa) . ki \ ^ f^a| 



L/2 

(79) 



In case the earth is assumed to be a perfect conductor, the equation reduces to 

H[)\{koaTa) = 0. 

The first two roots of the equation Hul{x) = are x = (2.38 + 0.17z) e~"', 
X = (5.6 + O.lTz) e""', so that the eigenvalue equation becomes 

(80) koar^ = (2.38 + 0.17^>"'^^ 



344 (S344) ELECTROMAGNETIC WAVES 

The integral for r can be easily approximated in certain cases. If 

»(r-r„)«l 
we may write /(r) -^ J f"{ro){r — VoY. Now 

where ^ is defined by the last equation, so that/(r) = {^/a^){r — TqY and 

(81) /boar. = k,a f ^2 {r - r,) dr = k,a{^ay4- - l)V2. 

Put, as before in equation (81) a = ro(l + d), then koGTa = J koa{^ay^^8^ and 
in case the earth is a perfect conductor, we must solve the equation: 

i(^ay~8^ = (A;oa)-'(2.38 + O.lTOe"". 

Note that ^ will depend on Vq and therefore also on 5. In case ^^^^ is a slowly 
varying function of 8, we may consider it a constant and then 

5 = (4.76 + OMi)'''e-''''\ko/^y'\koay\ 

Since i '^ X^^^ = k{ro)ro '^ koTo '^ koao(l — 8) we find that 

(82) t'^koa+ iko/^y'\koay'\4:.7Q + 0.34^ye"/^ 

Formula (82) shows that the imaginary part of t varies as {koay^^ for the 
case of a duct, as contrasted to the case of a homogeneous atmosphere where it 
varied as {koay^^. Since the attenuation of the field in decibels depends on 
the imaginary part of t, we can see that the range will be greatly increased in 
the case of a duct. For (kod) of the order of 10^, the range is multiplied by 4.6. 

12. The Electric Dipole 

The preceding work has been carried out for the case of the magnetic dipole 
because then the equations and the discontinuity conditions become simpler. 
However, the case of most practical interest is that of an electric dipole. In 
this section we shall give briefly the equations appropriate for an electric dipole. 

First, a new Hertz vector and potential must be defined. Instead of (2) 
we put 

(83) H = V X C-; 

where C\ again has only its radial component not zero so that Ci = (rUi , 0, 0). 
Instead of (6), we find 

(84) , A 1 ^ + 1 A i„ , ^ + ,V,. ,=,0 

dr € dr r sm 6 dd 66 



BERNARD FRIEDMAN 



(S345) 345 



and instead of (8), we have 



Ha, — 



de ' 



E, 



jiXdredr / 

(rU,), H. = He = E, 



er dr dd 



0. 



Since Ee and H^ must be continuous across a discontinuity surface r = constant, 
it follows that Ui and (1/e) (dUi/dr) must be continuous across this surface. 
Instead of (14) we have equation 



(86) 



1 djrun) , r 2 _ n{n + 1) 1 
€ dr [_ r^ J 



(rUn) ~ — 



d(r - h) 
2xr 



with the same boundary conditions as for (14). Put again v^ir) for those solu- 
tions of the homogeneous equation corresponding to (86), which are regular at 
the origin, and Wn\r) for those solutions of the homogeneous equation which 
satisfy the radiation condition at infinity. We find that, now, the Wronskian 
is no longer a constant, but that 

(87) W„{r)/er,{r) = constant. 

The formula for Ui(r, 6) is the same as (19): 

TT / «x 1 '^ vJr)Wn(h) 2n -f 1 „ , ^. ^ , 



(88) 






Vn(h)Wn{r) 2n + 1 



27r 



P„(cos e), 



r < h. 



The contour integral representation and all the transformations are unchanged. 
However, the eigenvalue problem becomes the following: 

Find those values of t for which there exists a solution of the differential 
equation, 

t{t + i) V , „ 



(89) 



^^im^(^j,^ 



dr e dr 



which is both regular at r = and satisfies the radiation condition (15) at 
infinity. 

Because of the different discontinuity conditions, the eigenvalue equation 
is changed. Instead of (41) we have 



■:{r) dr 



(rvt) 



rvt 



LA 

ei dr 



!r^'V..x(Ml 



'J.h(M 



so that the eigenvalue equation (44) becomes 



(90) 



{d/dr)(rwt,) 
rWi: 



e(a) 



i{k\ - ky'\ 



346 (S346) 



ELECTROMAGXETIC WAVES 



After transforming the contour integral for (88) and using the as\Tnptotic 
formulas for the Legendre functions, we get the following result, instead of (46 j 



(91) U{r, e) = - Z [^^^ 



l)i 



exp [ijt,- -i- i)d] w,Sh) y^tSf) 
TTsin^ J e(b) d (d/dr)(rw,.) \ aw,, (a) aw,, (a) 
eid) dt 



rw, 



Here t, , j = I, 2, • - • , are the eigenvalues of (89). 

Equation (89) can be treated by the W.K.B. method after a transformation 
.s made, to get rid of the e'^ factor. Put rv = e^'rv then equation (89) becomes 



(92) 



dr 



{rv) + [k 



tit + 1) 



(rv) = 



where k' = /r + h'^'W^'eY = k' - k {d' 'dr){k~'). This transformation is 
valid only if we assume e to be continuous and differentiable. Since we shall 
only apply the W.K.B. method to (89) for r > a, our assumption is justifiable. 
The eigenvalues t^ of (92) can be found in exactly the same way as we found 
the eigenvalues for (34) . We shall not repeat the work but one difference should 
be noted. In case the earth is assumed to be a perfect conductor, so ci = ki = oo , 
the eigenvalue equation (90) becomes 

(d/dr)irw,,) ^ ^ ^^ d_ 

rwt, r^a di' 



(rwt,) 



= 0, 



instead of riCt, = 0. 



Appendix I 



The Contour Integral on the Semicircle 

In this section we complete the discussion of the contour integral repre- 
sentation in (29) by showing that the integrals over the semi-circle in the first 
and second quadrant vanish as the radius of the circle goes to infinity. This 
justifies the expansion of the integral as a sum of residues and an integral over 
the imaginary axis. 

Consider, now. the integral 



(93) 



/ 



{t + i)v,{h)w,(r)P,{-cosd) 



W, 



sm wt 



dt 



over a quarter circle | ^ | = T in the first quadrant. Put t = Te'\ then from 
(40) we have 

I ^^(-cos d) j ^ exp {-r^sin 6} 
! sinTT^ - dT''- 



BERNARD FRIEDMAN (S347) 347 

To estimate the other factor in the integrand, we shall investigate the 
asymptotic behavior for large t of the solutions of the equation corresponding 

to (16): 



^^ + (,._M + i))(„).0. 



It can be easily verified by substitution in the differential equation that there 
are two independent solutions which for large t behave either like 



rv 



^■'•'"[i - 2FTT r ^^-^ *] °"- '' ~ ^^^"{i + k [ ^'^ *] 



where Ci and C2 are independent of r. Since rvt was by definition regular at 
r = 0, it must behave like Cjr'^^ for large t. When these asymptotic formulas 
are differentiated with respect to t, we find that for all solutions of the differential 
equations 



(94) 
Now 



{d/dr)( rv) 
rv 



r 



{t + i)Vt(b)w,{r) _ it + i)hvXb)rw,ir) 



hr\ rwt{r) — {rvt{r)) — rvt{r) — {rwt{r)) 



^ L+i [hvXh)]/[rvM] ^i^gj.^ ^ ^ (d/dt)irv,) _ id/dt){rw,) 
br M rVt rwt 

M will vanish if and only if Wt =0. li T is so chosen that the quarter circle 
does not go through any zero of Wt , then from (94) M will be bounded below 
by some multiple of t/r. We have then 

{t + h)v.{b)w,(T) _ ct (by ±_^c /5y "^ 

Wt '^ br \r/ t/r h \r) 

and since r < b, this term will go to zero as T goes to infinity. Finally, the 
integral will be less in absolute value than 



172/ exp l-Tdsin 6} Tdd {-) 



bST 

and as this clearly approaches zero for large T, the shifting of the contour in 
section 4 was justified. 



348 (S348) ELECTROMAGNETIC WAVES 

Appendix II 
An Estimate for the Integral 

It has been shown that the contour integral representation of the Hertz 
potential can be expressed (39) as a sum of residues plus an integral over the 
imaginary axis. In this section we estimate the value of the integral and show 
two things: 

First, the integral vanishes identically if the earth is assumed to be a perfect 
conductor; second, the integral is small compared to the first term in the residue 
series if the complex dielectric constant of the earth is small compared to the 
dielectric constant of the atmosphere. Since this condition is always satisfied 
in practice, we are justified in neglecting the contribution of the integral. 

In order to estimate the integrand in (39) we must make use of the fact 
that the differential equation (34) is unchanged when t is replaced by — ^ — 1 
so that 

wl'\r) = w[}U{r) and wl'\r) = w'^'l.,{r). 

Since wl^^ and w\^^ are independent solutions of (34), we may write 

v,{r) = awl'\r) + ^w[^\r) 

where a and /3 are independent of r but may depend on t. 

Note that Wt = ^ Wronskian {wl^\ wl^^) = 2iko(3. Now, we make use of 
the fact that for r < a, k^ = kl , is sl constant so that 

rvtir) = r^^^'Jt + ^^ihr), r < a. 

At the boundary r = a, we must have 

aawl'\d) 4- I3awl'\a) = a'^T.^.ik^a) 






+ ^|(™'™H =|('-"^-^-») 



These two equations will determine a and (3. We find 



™-'|(r-J,.j) 



J.^il(r^^n 



rwl" f^(.r-'J..,) 



J.Hf,(rwn 



(95) 



rwrir) I [log (r''V,.5)] - | [log (mi")] 
rwr(r) I [log (r"V,.j)] - | [log (m,"')] 






BERNARD FRIEDMAN 



(S349) 349 



where ilf , is defined by the last equation. Now, consider the expression 



2^/co 2ikowl'\a) 



w['\h)w['\r)M, . 



Everything in this formula, except the factor Mt , is unchanged when t is re- 
placed by — ^ — 1, so that 



Vt{h)wt{r) v^t-x{h)w-t-i(r) 



w?\a)w\'\h)w\'\r) 



Now 



w. 


• 


W^-,-1 


2ihw'i"{a) 


L^ 




M, = 


_|:log(r-j,.»)_ 


^-[|iog™-; 


o 




_! log (/-/,.»)_ 


^-[|iog™;-; 


a 



[M, - M_,_,]. 



(96) 



[|l^^-'^l-[l^^^-'^^^l 



[|log(r-/..)l-[|log..-] 



+ 1 



and we see that in this formula the only term which changes when t is replaced 
by — ^ — 1 is 



[log(r"V,..)L 



However, the argument of this Bessel function is k^a, which is a very large 
quantity since the conductivity of the earth is so large, and by the asymptotic 
formula for the Bessel functions of large argument we have 

[I log ('•"V„j)]^ = h tan {k,a - < | - |)- 

Note that this formula is valid only if k^a is very much larger than t. Now 
since k-^ is larger than k, the fraction in the formula (96) must be very small so 
that Mt — M_t-\ is nearly zero. 

If we put t equal to ir — J so that — ^ — 1 = — /r — J, we see that the 
bracket in the integrand of (39) must be nearly zero for r very much smaller 
than k-i_a. In the remaining interval of integration we use the formula (40). 
We find then that the integrand is of the order of magnitude e'^\ Now 



r -t9 1 1 -ka9 



350 (S350) ELECTROMAGNETIC WAVES 

and we see that this is very small compared to exp \ — (kay^^d] or to 
exp { — {kay^'^a] which is what is obtained from the residue series. We have 
thus shown that the integral in (39) is small compared to the first term in the 
residue series. 



Bibliography 

1. Watson, G. N., The diffraction of electrical waves by the ^arf /i,|Proceediiigs of the Royal 

Society of London, Volume A95, 1918-1919, p. 83. 

2. Bremmer, H., and van Der Pol, B., The diffraction of electro-magnetic waves from an elec- 

trical point source round a finitely conducting sphere, with applications to radio- 
telegraphy and the theory of the rainbow, The London, Edinburgh, and Dublin Philo- 
sophical Magazine and Journal of Science, Volume 24, 1937, p. f41. 

3. Rydbeck, Olof E. H., On the propagation of radio waves, Transactions Chalmers L^niversity 

of Technology, No. 34, 1944. 

4. Pryce, M. H., unpublished work. 

5. Booker, H. G., and Walkinshaw, W., The mode theory of tropospheric refraction and its 

relation to wave-guides and diffraction, Meteorological Factors in Radio Wave 
Proi>agation. Report of a conference held on April 8, 1946, by the Physical Society 
and The Royal Meteorological Society. The Physical Society, London, 1946, p. 80. 

6. Furry, W. H., Theory of characteristic functions in problems of anomalous propagation, 

O.E.M. Sr-262, Division 14, Report 680, Radiation Laboratory, :SI.I.T., Feb. 28, 
1945. 

7. Pekeris, C. L., Accuracy of the earth flattening approximation in the theory of microwave 

propagation, Report M.P.G.-3 Columbia University Mathematical Physics Group, 
April 15, 1946. 

8. Schelling, I. C, Burrows, C. R., and Ferrell, E. B., Ultra short wave propagation, Proceedings 

of the Institute of Radio Engineers, Volume 21, 1933, p. 427. 

9. Langer, R. E., On the asymptotic solutions of ordinary differential equations, with an applica- 

tion to the Bessel functions of large order. Transactions of the American Mathematical 
Society, Volume 33, 1931, p. 23. 

10. Eckersley, T. L., and Millington, G., Application of the phase integral method to the analysis 

of the diffraction and refraction of wireless waves round the earth. Philosophical Trans- 
actions of the Royal Society of London, Volume A237, 1938, p. 273. 

11. Bremmer, H., Terrestrial Radio Waves. Elsevier Publishing Company, Xew York- 

Houston, 1949. 

12. Stratton, J. A., Electromagnetic Theory, McGraw-Hill, New York, 1941, p. 415. Frank, P., 

and von Mises, R., Die Differential und Integralgleichungen der Mechanik und 
Physik, Vol. II, Vieweg, Braunschweig, 1930, p. 871. 

13. Stratton, .1. A., Electromagnetic Theory, McGraw-Hill, New York, 1941, p. 431. 

14. Hobson, E. W., The Theory of Spherical and Ellipsoidal Harmonics, University Press, 

Cambridge, 1931. 

15. Pekeris, C. L., Wave theoretical interpretation of propagation of 10-centi meter and 3-centimeter 

waves in low-level ocean ducts, Proceedings of the Institute of Radio Engineers, 
Volume 35, 1947, p. 453. 

16. Langer, R. E., The asymptotic solutions of certain linear ordinary differential equations of the 

second order. Transactions of the American Mathematical Society, Volume 36, 
1934, p. 90. 



Reflection of Electromagnetic Waves from 
Slightly Rough Surfaces 



By STEPHEN O. RICE 
Bell Telephone Laboratories 

I. Intraduction 

The problem of reflection of waves from a rough or corrugated surface is 
of interest in a number of different fields. In particular, the problem occurs in 
the propagation of radio Avaves over rough ground or over the sea. The general 
problem of reflection from rough surfaces appears to be difficult. However, a 
number of investigators have been able to make progress by dealing with special 
cases or by making suitable assumptions. For example, the radio problem in 
which the impressed field originates at a point has been studied by E. Feinberg 
[1]. L. V. Blake [2] has applied probability theory to the problem of calculating 
the reflection of radio waves from a rough sea. The reflection from the very 
rough surface formed by the edges of an infinite set of plates has been investi- 
gated by J. F. Carlson and A. E. Heins [3]. Also, in addition to the studies of 
W. S. Ament^ mentioned below, I understand that a considerable amount of 
work on this subject, which is unpubhshed as yet, has been done by Mr. Twersky 
[7] of New York University, and by Messrs. Norton and Hufford and their 
associates at the National Bureau of Standards. 

Here we shall be concerned with the reflection of plane electromagnetic 
waves from a surface z = f{x, y) which is almost, but not quite, flat. The 
small deviations of this surface from the x,?/-plane are of a random nature. 
Except in Section 7, the surface is assumed to be a perfect conductor. Although 
in practical cases the surfaces are usually much rougher than is assumed here, 
our problem has the virtue of being one of the simplest which still shows the 
effect of roughness. 

The roughness of the surface is described by a ' 'roughness spectrum" or 
' 'roughness distribution function" TF(p, g). When the surface is expressed as 

Paper presented at the June, 1950, Symposium on the Theory of Electromagnetic Waves, under 
the sponsorship of the Washington Square College of Arts and Science and the Institute for 
Mathematics and Mechanics of New York University and the Geoph3\sicaI Research Direc- 
torate of the Air Force Cambridge Research Laboratories. 

^I am indebted to Mr. Ament of the Naval Research Laboratory for an opportunity to 
study some of his work before its publication. I also wish to acknowledge the help I have 
received from discussions of the general problem of reflection with K. BuUington of the Bell 
Telephone Laboratories. 

351 (S351) 



352 (S352) ELECTROMAGNETIC WAVES 

the sum of two-dimensional Fourier components, W(p, q) dp dq represents the 
relative strength (as measured by their contribution to the mean square value 
of /(.r, y)) of those components which go through between p and p -{- dp radians/ 
meter in the x direction and through between q and q -{- dq in the y direction. 
If the average distance between the hills on the surface is large, and the surface 
is smooth except for these hiQs, W(p, q) will be appreciably different from zero 
only for small values of p and q. The associated auto-correlation function of 
the surface, which may be expressed as the Fourier transform of W(p, q), is 
not used here although it does occur in the work of W. S. Ament. 

The reflected field is determined by a method similar to that used by 
Rayleigh [4] to study the reflection of acoustic waves from rough walls. The 
expressions which we obtain for the field are not exact since the boundarj^ condi- 
tions at the surface are satisfied only to wdthin 0(f{x, y)). i.e. to ^-ithin terms 
of the second order — a shortcomiQg forced upon us by the increasing complexity 
of our successive approximations. The two cases corresponding to horizontal 
polarization (incident E vector parallel to x,7/-plane) and vertical polarization 
(incident H vector parallel to a;,t/-plane), respectively, are considered. 

After expressions for the components of the field are obtained, various 
averages are computed, the average being taken over many surfaces which are 
different but which have the same statistical properties. In particular, the 
average value of the reflected field leads to an expression for the reflection co- 
efficient. It is found that this reflection coeflficient depends upon the polariza- 
tion in somewhat the same way as does the reflection coeflBcient for an almost, 
but not quite, perfectly conducting plane. Also, when the average distance 
between hills is large, the reflection coeflficients for both the horizontal and 
vertical polarizations reduce to the same expression. By a method similar to 
the one used in the study of Fraunhofer diffractions, W. S. Ament has obtained 
an expression for the average reflection coefficient when the distance between 
hills is large and they are such that they do not cast any shadows (with respect 
to the incident wave). Our approximate expression agrees with the first two 
terms in the expansion of Ament 's expression, which is as much of an agreement 
as the accuracy of our work allows. 

Closely associated with the problem of reflection is the problem of surface 
wave propagation. This corresponds loosely to the case of grazing incidence 
and vertical poki^-ization; a modified form of the reflection analysis may be used 
to obtain an expression for the propagation constant of the surface wave. It 
is found that, roughly speaking, the Fourier components of the surface whose 
wavelengths are much greater than that of the electromagnetic wave tend to 
produce attenuation through scattering, while the guiding action of the surface 
is due to the components of shorter wavelength. This is in accord vrith the 
results of earlier studies of surface waves on corrugated surfaces [5, 6]. 

The method used to study reflection from a slightly rough but perfectly 
conducting surface may be extended to take into account the electrical prop- 
erties of the reflecting medium. This is done in Section 7 for the case of hori- 



STEPHEN O. RICE 



(S353) 353 



zontal polarization, the magnetic permeabilities of the two media being assumed 
equal. There are two reasons for this study. The first is to determine the 
additional amount of complication introduced. The second is to show that an 
annoying difficulty encountered in the perfect conductor case, namely that the 
integral for the mean square value of E^ (i.e. the component of electric intensity 
which is approximately normal to the surface) sometimes diverges logarithmic- 
ally, may be removed by taking into account the finite conductivity of the 
reflector. 

2. Description of Rough Surface 

We shall take the equation of the perfectly conducting rough surface to be 
2 = /(^j y) = Yj Pi'^) ^) exp { —ia{mx + ny)} 



(2.1) 



a = 2'k/L 



where the double summation extends from — qd to + °° for both m and n. 
The definition of a shows that f{x, y) is periodic in both a; and 2/ with period L 
(assumed to be large) . In order to make f{x, y) real we impose the condition 

(2.2) P{-m, -n) = P*{m, n) 

where the asterisk denotes the conjugate complex quantity. 



INCIDENT E FOR 7 

VERTICAL POLARIZATION 



MAI N REFLECTED RAY 
g-i,^ (QfX + JTz) 



SLIGHTLY ROUGH SURFACE 

z=fCx,y) 




Fig. 1. Diagram showing the coordinate system and the incident E vector for vertical 
polarization. For horizontal polarization the incident E vector is parallel to the y axis. 

The random character of roughness is introduced by taking the coefficients 
P(w, n) to be independent random variables, subject only to (2.2). For the 
sake of being definite, we assume P(0, 0) and the real and imaginary parts of 
P(0, 1), P(l, 0), P(2, 0), P(l, 1), P(0, 2), P(l, -1), etc. to be independent 
random variables distributed normally about zero. We assume further that, for 
assigned values of m and n, the four independent random variables formed by 
the real and imaginary parts of P{m, n) and P{m, —n) all have the same vari- 



354 (S354) ELECTROMAGNETIC WAVES 

ance, i.e., the same mean square value. When we use angular brackets to denote 
average values, our assumptions tell us that 

(P(m, n)) = 
(P(m, n)P(u, v)) = 0, (u, v) 9^ (— m, —n) 
(2.3) (P(m, n)P*(m, n)) = (P(m, n)P{-m, -n)) = ir'Wip, q)/L' 

W{p,q) = W{\p\,\q\) 

p = am = 2'jrm/L, q = an = 2Tm/L 

Here ( ) denotes that m and n are to be held fixed and the average taken 
over the universes of the real and imaginary parts of the P(m, 7?)'s. W{p, q) is 
the roughness spectrum mentioned in the introduction, and p and q are radian 
wave numbers. Note that {P^im, n)) is zero, except when m = ?? = 0, by 
virtue of the real and imaginary parts of P{m, n) having the same variance. 
Incidentally, the statistical properties of P(m, n) were obtained by expressing 
the typical Fourier series term 

{amn COS amx + 6^„ sin amx) cos any 
(2.4) 

+ {Cmn cos amx + d„^n sin amx) sin any, m > 0, n > 

as the sum of four exponential terms. This leads to four relations of the form 

P{m, n) = 

and the properties of P(m, n) follow when a«„ , * • • , c?„n are assumed to be 
independent random variables distributed normally about zero T\'ith the same 
variance, namely ^-w^Wip, q)/L^. The 47r^ arises from the fact that we have 
elected to measure p and q in radians/meter instead of cycles/meter. 

Equation (2.1) defines a surface for each set of coefficients. As an example 
of the use we shall make of TF(p, q) we compute the average value of f{x, y) 
as we go from surface to surface, holding x and y fixed all the while. 



(2.5) 



(/'(*) y)) — 2 {Pi'm, n)P{u, v)} exp { —ittx(m + ii) — iay{n + v) 

mnuv 

= Y.{Pi.^,n)P{-m, -n)) 
-> [ dm \ dn ir'Wip, q)/V 



STEPHEN O. RICE (S355) 355 

Here we have used (2.3). In going from the summation to the double integral 
we have let the period L approach infinity. It is seen that Wij), q) dp dq/4: 
represents the contribution to {f(x, y)) of those components in (2.1) having 
between p and p -\- dp radians/meter in the x direction and between q and 
q -]r dq radians/meter in the y direction. 

3. Incident Wave Horizontally Polarized 

We assiune the components of the total electric intensity for z > f{x, y) 
to be 

E, = Yj A^nE{m, n, z) 

(3.1) Ey = 2i sin ^yz exp { —iavx] + 2Z BmnE(m, n, z) 

E. = J2 CmnE{m, n, z) 
where the summations extend from — oo to -}- oo for m and n and 

(3.2) E{m, n, z) = exp {—ia{mx + ny) — ib{m ,n)z]. 

The time factor exp [ioit] is understood. 6(m, n) is either a positive real or a 
negative imaginary number: 

( [/3^ - a'm' - aV]'/^ m' -\- n' < /37a' 

(3.3) h{m, 71} = \ 

\-i\a-m' + aV - ^^\''\ m' ^ n' > ^'/o: 

where ^ = 27r/X, X being the wavelength of the incident wave. A^n , B^n , C^n 
are constants which we shall determine approximately on the assumption that 
|8/ and the partial derivatives /;, and fy are small (here and in what follows we 
shall often denote f(x, y) by /) compared to unity. 

The field obtained from (3.1) when the simamations are omitted is the one 
which w^ould occur if the perfectly conducting surface were flat (/ = 0). In 
(3.1) we take v to be an integer so that the field is periodic in x and y of period 
L by virtue of a = 27r/L. It follows that the angle 6 between the incoming 
ray and the 2;-axis is restricted to certain discrete values given by 

av = 2ivv/L = 13 sin d = 13a \ av \ <^ 
(3.4) 

a = sin 6, y = cos 6 < 7. 

Since L becomes very large we can pick an integer y which will correspond 
approximately to any angle of incidence. The leading term in Ey may be 
written as 

exp {—i^{ax — yz)] —exp {—i(3{ax + 7^)t 

where the first term represents the incoming wave and the second term the 



356 (S356) ELECTROMAGNETIC WAVES 

main part of the reflected wave. It is seen that the direction cosines of the 
incident and reflected rays are {a, 0, —y) and (a, 0, y), respective!}'. From 
the definition of h{m, n) it follows that 

(3.6) Hv, 0) = (/3^ - aVf" = 0y. 

The exponential form (3.2) of E(m, n, z) ensures that all three components 
of the electric intensity (3.1) satisfy the wave equation. The coefficients are 
determined by the relation div E = 0, which gives 

(3.6) amA^^ + anB^, + h(m, n)Cran = 0, 

together with the condition that the tangential component of E must vanish 
at the perfectly conducting surface z = f. If N denotes the unit vector normal 
to the surface, N(E'N) is the component of the electric intensity normal to the 
surface. The remaining portion of E, the tangential component, isE — X(E-X), 
all three components of which must vanish. Equating the x and y components 
to zero gives 

E^ - NAE^N^, + E,N, + E^\) = 
13.7) 

E, - N,{E,N, + E,N, + ^,.YJ = - 

If these two equations are satisfied the z component is also zero (if X, ^ 0) as 
may be seen by multiplying the first by N^ , the second by Ny , and adding. 
The components of N are 

(3.8) N^ = -J.N. , N, = -f,N. , N. = (1 + /; + /;)-^''^ 

We now assume /3/, /^ , fy all to be of the same order of smaUness wliich, 
for the sake of simplicity, we shall denote bj^ 0(/) instead of 0(/3f). Likewise 
instead of O(^^f) we shall write 0(f), and so on. In our work we shall neglect 
0{f) terms and it will not be necessary to go beyond the leading tenns in 

(3.9) N. = -/. + Oif), Ny = -fy + Oif), X. = 1 + 0(f) 

Near the surface z — f, i.e. near ^ = 0, the leading term in Ey as given by 
(3.1) is 0(f), and we assume for the moment that E^ and E^ are also of this 
order. Then, neglecting O(f^) terms in (3.7), we obtain the two boundaiy 
conditions 

E, - X^E, = 
(3.10) 

Ey - XyE: = 

which must hold at 2; == /. Thus, if E, is 0(f), then both E, and Ey must be 
O(f^) at the surface. Of course this holds only for horizontal polarization. 
For vertical polarization it turns out that E, is 0(1) and both E^ and Ey are 0(f). 
The problem now is to choose the coefficients in (3.1) so that the divergence 
relation (3.6) and the boundary conditions (3.10) are satisfied to within 0(/"). 



STEPHEN O. RICE 



(S357) 357 



Writing 

sin ^7/ = /37/ + Oyf) 

(3.11) E{m, n, f) = [1 - ih(m, n)f + • ■ -Mm, n, 0) 

A = /I (1) 4- i ^2) 4- . .. 

where A^Jli is 0(/), A^^i is 0(/^), etc., and expressing B^^ , Cmn in a similar way 
enables us to write the boundary conditions (3.10) as 

J2 [A^j;: + AZ + f.C'^l][l - ib{m, n)/Mm, n, 0) = 

(3.12) 2i exp { -iavx} -^yf 

+ E [5^:1 + B^^i + f.C'llm - ib{m, n)f]E{m, n, 0) = 

where we have neglected 0(f) terms. In this work we shall overlook questions 
of convergence although they may perhaps be treated by placing suitable re- 
strictions on the components P(m, n) of the surface/. 
Equating the first order terms in (3.12) to zero gives 

(3.13) 

2i exp { -iavx]^yj + X) B^l\E{m, n, 0) = 0. 

Lil^ewise, the second order terms yield 

Z UZ + f.C^ll - ih(m, n)fA'li]E(m, n, 0) = 
(3.14) 

E [BZi + LC'll - ih{m, n)fB'il]E{m, n, 0) = 0. 

As (3.2) shows, E(m, n, 0) is the exponential function of x and y which occurs 
in a double Fourier series. Hence the first of equations (3.13) requires A ^Jii = 0. 
In order to interpret the remaining equations we need the following results. 
Writing u, v for m, n in (2.1) and using the definition of E(m, n, z) leads to 



(3.15) 



"/■ 




1 


/x 


= E 


— iau 


-L. 




_ —iav_ 



P{u, v)E{u, V, 0) 



whence, upon setting m = u -\- v, n = v, 

exp { —iavx]f = E ^&*j v)E{u -}- v, v, 0) 

(3.16) 

= E Pi'^ — ^, n)E{m, n, 0) . 

mn 

A somewhat similar argument may be used to establish 



358 (S358) 



ELECTROMAGNETIC WAVES 



(3.17) 



If. 



J„nE{m, n, 0) 



= Z 



-ia{m — k) 
_ — ia{n — I) , 



JuP{rn — k,n — l)E{m, n, 0) 



where the summation for m, n, k, I on the right extends from — co to oo and 
J„y, represents an arbitrars^ function of m and n. (3.17) is obtained by replacing 
m, n by k, I on the left and then introducing (3.15). The two E functions may 
be combined by the multipUcation law for the exponential function and the 
right side of (3.17) obtained upon setting m = u -^ k, n = v -\- I. 

Equating the coefficient of E{m, n, 0) to zero in the second of equations 
(3.13) after using (3.16) gives 



(3.18) 



B 



-2i^yP{m - v,n). 



The second order terms A^l^^, B^l may be obtained by setting the values of 
A^^:, 5^J.: in (3.14) and using (3.17): 



(3.19) 



ATn= Z iair^^ - h)ClVPim - k, n - I) 

ki 

B^l = E Mn - Odr + 20yh(k, l)P(k - v, l)]P(m - k, n - I). 



Once the A's and 5's are knovn the C^s may be obtained from the divergence 
relation (3.6). For example 

(3.20) CLV, = -anB'l\/h{m, n) = 2i^yanP(m - v, n)/h{77i, n). 

^Tien the appropriate expressions for the coefficients are set in (3.1) we get 

E, = -2/37 Z E{7n, n,z) Z a'i^^ - k)lQ{m, n, k, I) 



E, = 2i exp {-il3ax] sin ^yz - 2/37 S E{m, n, z)[iP{ 



m 



n) 



(3.21) + E [a\n - 1)1 - l\k, l)\Q{m, n, k, I)] 

kl 

E, = 2/37 S lE(m, n, z)/h{m, n)][ianP{m - v, n) 

mn 

+ X WKm + n^ - mk - nl) - anh\k, l)]Q{m, n, k, /)] 



STEPHEN O. RICE (S359) 359 

where E{m, n, z) is the exponential function defined by (3.2), (a, 0, —7) are 
direction cosines of the incident ray, a — 2'k/L where L is the period of the 
surface, v is an integer given by av = jSa, /3 = 2t/\ and 

(3.22) Q{m, n, k, I) = P{k - v, T)P{m - k, n - l)/b{k, T) 

The summations for m, n, k, I extend from — 00 to + <» . 

The terms entering the summations in (3.21) may be divided into two 
classes. A term is in the first class if the corresponding values of m and n satisfy 
a^m^ + a^n^ < 0^ and in the second if the opposite inequality is satisfied. For 
a term in the first class h{m, n) is positive real and E{m, n, z) represents a wave 
traveling in the direction specified by the direction cosines am/^, an/^, h{m, n)/^. 
The corresponding electric intensity is perpendicular to the direction of propaga- 
tion, as is shown by (3.6). In terms of the wavelength X of the incident wave 
and the period L of the surface these direction cosines are Xm/L, \n/L, b(m, n)/^. 
For a term in the second class 6(m, n) is negative imaginary and E{m, n, z) 
corresponds to a surface wave traveling in the direction determined bj^ m/n and 
exponentially attenuated in the z direction. 

An examination of the series for E, shows that the terms become large for 
n near zb/S/a = d=L/\ and for m near zero if the coefficients around P{ — v, /5/a) 
are appreciably different from zero, for then h{m, n) in the denominator is small. 
This indicates that for some surfaces there Avill be an appreciable sidewise 
(i.e., in the y direction) scattering of the wave. It will be seen later that if the 
finite conductivity of the reflecting surface is taken into account the large 
terms remain finite even if 6(m, n) = 0. 

That E^ sometimes tends to be large may be seen from the following physical 
considerations. Take the case of normal incidence so that ^^ = and take the 
surface to be 2; = 2P cos ^y. The incident Ey produces a surface current in 
the y direction and each upward (and downward) slope of the surface may be 
regarded as a surface current element (infinitely long in the x direction) which 
radiates a field. Since the period of the surface is equal to one wavelength at 
the incident radiation, the E^ components of the fields of various current ele- 
ments are in phase at 2; = and hence the resultant E^, tends to be large. 

As the roughness increases, the additional energy in the scattered radiation 
is obtained at the expense of the energy in the main component of the reflected 
wave. This is closely connected with the relation 

(3.23) Real Part of 2^yB,, = Z [| ^L | + I ^L | + I CL \]h{m, n) 

which is an extension of a result due to Rayleigh [4]. Here the summation 
extends over all values of the integers m and n such that rn -\- n^ < ^^ ja (i.e., 
over the values for which 6(m, n) is real). Equation (3.23) is an exact relation 
and does not depend on 2 = j{z, y) being only slightly rough. It may be es- 
tablished by equating to zero the average power flow through a square of side L 
lying on a plane z = constant parallel to the a;,2/-plane and at a great height 
above it. B,q is the change in the main reflected wave produced by the rough- 



360 (S360) ELECTROMAGNETIC WAVES 

ness. iilthough (3.23) was derived by integrating Poynting's vector over the 
square, it is interesting to note that the m,n-th term on the right is proportional 
to the intensity of the m,n-th component of the field times the cosine, 5(m, n)/l3, 
of the angle between its direction of propagation and the z-axis. That (3.23) 
and (3.21) are in accord may be seen from the fact that (since iP{v — v, 0) is 
imaginary) the real part of B^o is, from (3.21) and (3.1), 

(3.24) 2/37 Z [o"l" + h'ik, I)] \ P{k - v, I) \Vh{k, I) + 0(f) 

where the summation extends over values of k and I such that k^ -{• f < ^ jci 
(because 6 (A;, I) is real for only these values). Furthermore, 

I A.„ h^ = Oif), I B^^ r = I 2/57P(m - ., n) T + 0(f) 

I C.. I' = I 2^yanP{m - v, n)/5(m, n) \' + 0(f) 

and when these are put in the right hand side of (3.23) we get a result which 
agrees mth (3.24). 

Up to this point the results of this section hold for any assigned values of 
the P{m, n)^s except that they are usually required to be small. No statistical 
considerations enter into equations (3.1) to (3.21). However, from here to the 
end of this section we shall make use of the statistical properties of the P(w, n)'s, 
described in Section 2, to obtain various average values from the approximate 
expressions (3.21) for the field. From (2.3) and (3.22) follows 

( 0, (m, n) ^ (v, 0) 

(3.25) {Qim,n,k,l)} = < 

WW(ak - av, al)/L%{k, I), m = v,n = 0. 

When the averages of E^ , Ey and E^ as given by (3.21) are taken only the terms 
for which m = v, n = remain. Furthermore, since the first power of Z is a 
factor of the terms remaining in E^ and E^ and since TT'" and h{k, I) are even 
functions of I, it may be shown that the average values of E^ and E^ vanish. 
This is to be expected on ph3^sical grounds. 
The average value of j^^ is 

{Ey) = 2i exp {—i^ax} sin /Jyz 

+ 20yE(v, 0, z) S [^1^ + b{k, o] ^_ W{ak - av, T) 

-^ exp {—i^iax — yz)} 

- exp 1 -,/3(a. + T.) ijl - 2/3 £ dr £ <fe [f + ^b] M?^^^^} 
where we have used av = ^a and have set 



(3.26) 



(3.27) 



STEPHEN O. RICE 

r = ak = 2Trk/L, s = al 



(S361) 361 



i[r 



' ' s' 



^r\ 



< r^ + s 



In going from the summation to the integration we have assumed L to approach 
infinity just as in (2.5). By setting 

y = r ~ (3a, q = s 



(3.28) j [^2 _ (p + ^^y- _ qY'- 

l-iKp + ^af + q' - 13']' 
the last term in (3.26) may be written as 



(3.29) 



/OO /»0O 

dp I dq 



yq 



+ yh 



W{v, q) 
4 ' 



The coefficient of exp [—i^{ax + yz)] in (3.26) represents the average 
vakie of the reflection coefficient and hence (3.29) represents the change in the 
reflection coefficient produced by the roughness. 

The leading term in the mean square value of the fluctuation of Ey about 
the value it has in the absence of roughness is 

(I Ey - 2i exp { -i^ax] sin ^yz |') 

= 4.(3S' Z E*{m, n, z)E{k, I, z){P*{m - v, n)P(k - v, I)) 



(3.30) 



where 



(3.31) 



= 4/3^7' E exp [-z<p{k, l)]Tr'W{ak - av, al)/L' 
ki 

-^ 4:(3y j dr J ds e-'^Wir - ^a, s)/4 
= 4/5V f dp [ dqe-nVip, g)/4 

»/ — 00 *^ — OO 

<pik, I) = ih{k, I) - ih*{k, I) 

= Imaginary part of —2h{k, I) 

i 0, k' ^f < /3^/a^ 

b[aT + aH' - (3T", k' +f> ^'/a' 



362 (S362) ELECTROMAGNETIC WAVES 

and V? = when r' + s' < /3^ or (p + /3a)' -]- q^ < ^'' and 

(3.32) ^ = 2[r' + ^^ - ^r' = 2[(p + ^a)' + g' - /Sl^^^ 

when the inequaUties are reversed. It is interestmg to note that the average 
value of 

Ey — 2^■ exp { — i^ax ] sin ^yz 

is O(^^f) (this is indicated by (3.26) and (3.29) since the double integral of 
W(p, q) gives (,f{x, y))) while the rms value of its modulus, as obtained by 
the square root of (3.30), is OW). 

When the procedure used to derive (3.30) is applied to the 0(/) terms in 
the expressions for E^ and E'^ in (3.21) we obtain 

(3.33) •-" •-" 



4161 



(I E, D = 0. 
where, from (3.28), 
(3.34) \b\' = \0' -{p + 0aY - q' \. 

Here we encounter trouble because the denominator may become zero. If 
W{pj q) is continuous and not zero on the circle | 6 |^ = in the p, q plane, the 
double integral in (3.33) diverges logarithmically. This difficulty may be over- 
come in several ways. If the reflecting surface is not perfecth' conducting a 
convergent double integral analogous to (3.33) may be obtained from the ex- 
pression for C^li given by (7.21). When the conductivity g of the reflecting 
surface is large, but not infinite, equation (7.26) shows that the h occurring in 
the I 6 1^ of the denominator of (3.33) should be replaced by b + i3^/r where r 
is the intrinsic propagation constant of the reflecting material: r = (^iccngY^', it 
being assumed that the permeability /x (/; = 47r x 10~' henries/meter for free 
space) is the same for the reflecting material as for the region z > f{x, y). Since 
h is purely real or imaginary, it is seen that the new denominator never vanishes. 
Another method of meeting the difficulty is to assume /(x, ^) = outside a 
square of side L instead of taking it to be a periodic function. The integral vriW 
converge as long as L remains finite. 

Equations (3.30) and (3.33) show that the first approximation to the scat- 
tered field vanishes as z approaches infinity if W{p, q) is zero for the region 
inside the circle (p + ^aY -^ q = ^ where <^ is zero, i.e., if the average distance 
between the hills is rather small compared to a wavelength. This means that 
the reflection in this case is perfect (the modulus of the average reflection co- 
efficient being unity). 

Incidentally, as Rajdeigh has pointed out, the reflection from a simple 
sine wave surface will be perfect if the period of the sine wave is small enough. 



STEPHEN O. RICE (S363) 363 

In order to see this from our analysis suppose the equation of the surface to be 
z = 2P cos {rriiax + niay) so that all of the P{m, nYs are zero except 

P(mi , n,) = P(— mi , — Til) = P = real. 

The only non-vanishing 0(f) terms in (3.21) are the two given by m = v zk rrii , 
n = dzTii (e.g., the upper signs go together), and the only non-vanishing 0(f) 
terms are the three given by m = v zt 2mi , n = zb2ni and m = v, n = 0. It 
follows that if nii and rii are such that 

(3.35) (1 H - I Jni I)' + «! > /37a' 

the only term which can possibly correspond to a scattered wave is the one 
given by m = V, n = (remember that | ^ | < ^/a) because all of the others 
correspond to surface waves which carry no energy away from the surface. 
Since the m = v, n = term corresponds to a wave traveling in the same direc- 
tion as the main reflected wave it cannot be regarded as scattering. All it can 
do is change the phase of the reflection coefficient. Our work doesn't go beyond 
0(f) terms but it doesn't seem likely that the higher order approximations will 
bring in any terms which can be interpreted as scattering. 

However, the situation is quite different if the surface consists of the sum 
of two (or more) rapidly varying sine waves whose "interference pattern" has 
a period long enough to produce scattering. For example, let the surface be 

(3.36) z = 2Pi cos (niiax + riiay) + 2P2 cos (nizax + n2ay) 

where nii , rii satisfy (3.35) and mg , ng satisfy a similar inequality. An examina- 
tion of (3.21) and the definition of Q(m, n, k, I) shows that the 0(f) terms 
which might produce scattering are the two for which m = v zt (rrii — ma), 
n = =b(ni — rig). At least one of these is certain to produce scattering if 

(3.37) (I H - I ^1 - ^2 I)' + (n, - n,Y < ^'/a\ 

because it would correspond to a wave for which him, n) is real and hence would 
carry energy away from the surface in a direction different from that of the 
main reflected wave. Even if (3.37) were not satisfied there is a possibility of 
higher order terms corresponding to scattering. 

If we now consider the case of the rough surface with the above examples 
in mind we see that although the reflection may sometimes be perfect to a first 
approximation, the 0(f) terms in (3.21) give rise to a scattered field (somewhat 
similar to the Rayleigh scattering produced by small particles) which does not 
vanish as z becomes large. In order to study mean square values involving the 
0(f) terms it is necessary to deal with averages of expressions containing the 
product of four P(7n, n)'s. Since the results appear to be rather complicated, we 
shall not go farther than to state the following result which may be applied to our 
problem when P„ is replaced by P(m, n) and the summation taken with respect 
to m, n instead of w, and likewise for k, n' , k' . 

Let Po be real. Let Pq and the real and imaginaiy parts of Pi , P2 , • • • 



364 (S364) ELECTROMAGNETIC WAVES 

be independent random variables with average value zero. Let the real and 
imaginary parts oi P^ , n > 0, have the same mean square value so that (Pi) = 
unless 71 = 0, and define P_„ as the conjugate complex of Pr, so that 

{p_.p„) = (I P. D = (I Pl I), 

(P^P,:) = if m9^ -n.^ 

If F{n, k, n', k') denotes an arbitrarj^ function of n, k, n\ k' it can be shown 
that, if the summations run from — co to + ^ , 

E F{n, k, 7^^ k'){PnP,Pn'P,-) 
nkn'k' 

= Z \F(n, -n, k, -k) + F(n, k, -n, -k) + F{n, k, -k, -n)] 

nk 

(3.38) ■{] p:\x\pi\) 

+ S [^(^j "~^; ^^; "~^0 + ^(^j ^j ~^j ~^) + F{n, —n, —n, n)] 

n 

.l{\Pt\)-2{{\Pl\}y-^ 

+ F(o, 0, 0, o)mpi)r ~ 2(Pt}]. 

One method of establishing this result is to break the four-fold summation 
into the subgroups for which (1) k ^ 7i, k 9^ —n, (2) k = n^ n 9^ 0, (3) k = —n, 
n 9^ 0, (4) A; = 0, ?z = 0. The terms which have averages different from zero 
in subgroup (1) are those for which (la) 71' = —7i,k' = —k, (lb) n' = —A:, 
k' = —n. Likewise for the other groups we have (2a) n' = k' = —71, (3a) 
n' = -ti, k' = n, (3b) n' = n, k' = -n, (3c) n' = -A;' but ti' 9^ ±n, (4a) 
n' = k' = 0, (4b) n' = -k', n' 9^ 0. 

When, as in the case of the rough surface, the surface z = f(x, y) has many 
Fourier components of the same order of magnitude, the onh' term of importance 
on the right hand side of (3.38) is the double summation over 71 and k. This 
term goes into a fourfold integral invohdng the product of two W{jp, q) functions. 

4. Incident Wave Vertically Polarized 

In this section we assume the electric intensity of the field to be, in the 
absence of roughness. 

El = 2iy exp { —i^ax] sin ^yz, El = 
(4.1) 

E^ = 2a exp { — i^ax j cos ^yz 

where the symbols have the same meaning as in Section 3. In particular. 



(4.2) 



STEPHEN O. RICE (S365) 365 

^ = 27r/X, and (a, 0, —7), (a, 0, 7) are the direction cosines of the incident and 
reflected rays, respectively. A procedure similar to that used to obtain (3.21) 
leads to the following expressions, accurate to 0(f) terms, for the electric in- 
tensity in the presence of the slightly rough surface. 

E, = E^, -{- 2 J^ E{m, n, z)[i{aam - 0)P{m - v, n) 

tnn 

+ X) !a'(w - k){v - k)^ + (/3 - aam)h\k, l)]Q{m, n, k, I)] 
ki 

Ey = 2a ^ E{m, n, z)[ianP(m — v, n) 

mn 

+ Z !«(^ - 0(^ - k)^ - anh\k, l)]Q(m, n, k, I)] 

ki 

E,= E: + 2J2 [E{m, n, z)/h{m, n)] 

mn 

• i{a{m — p)^ + ah^im, n)]P(m — v, n) 

+ Z W{k - ^)(w' -\-n^ - mk - nW 

ki 

+ a[aa{m -f n^) — m^]h^{k, l)}Q(m, n, k, I) \. 

In these equations E(m, n, z) is the exponential function of x, y, z defined 
by (3.2) and Q{m, n, k, I) is the function (3.22) containing the product of two P's. 

The average electric intensity of the reflected wave is in the direction 
specified by the direction cosines (—7, 0, a). The corresponding wave function 
approaches, as L -^00, E{v, 0, z) multiplied by 

1 - 2^ J dr j ds ULJ^^ + 76jTF(r - /3a, s)/4 
= 1-2/3 f^ dp f^ dq [^ + yhjWip, q)/4. 



(4.3) 



where h is defined as a function of r, s and p, q by equations (3.27) and (3.28). 
The derivation of (4.3) is similar to that of its analogue in (3.26): the expressions 
(4.2) are averaged, the reflected wave picked out, and the square root of the 
sum of the squares of its x, y, z components taken (the average y component 
turns out to be zero). 

An idea of how the field components vary about their values in the absence 
of roughness may be obtained from the following analogues of (3.30) and (3.33). 



366 (S366) ELECTROMAGNETIC WAVES 

(I E, - E: r) = 4 r dp r dqe-''(ap - ffyrWip, q)/4. 

J —CO J —03 

(4.4) (I E, r> = 4 ]"" dp I" dq e-'cc'q'Wip, q)/4: 

(I E, - E: \'} = 4 r dp r dqe"%pl3 + ab'fWip, q)/4 \ b \\ 

— 00 «' — 00 

Here, as in (3.33), the last integral may not converge on the chcle 6 = 0. It 
was pointed out that this difficulty can be overcome in the case of horizontal 
polarization by considering the electrical properties of the reflecting surface, 
and the same is probably also true for the case of vertical polarization. 
The analogue of (3.23) is 

(4.5) Real Part of 2^A^o = Z [| ^L 1+ | 5L i + | CL Mm, n) 

where the summation extends only over those values of m and n for which 
him, n) is real {in + 7i < /3^/a^) and Amn etc. a're defined by 

E. = C + Z A^Mm, n, z) 
E, = H B^,,E(m, n, z) 

E.=E:+ J2C^Mm,n,z). 

Equation (4.5) and C^o = ~«^.o/7, which follows from the divergence relation 
(3.6), give a partial check on equations (4.2). 

5. Special Cases 

Suppose that the roughness spectriun, W(p, q) is zero except for a small 
region around p = 0, q = 0. In this case the average distance between the hills 
of the surface is large compared to the wavelength of the incident radiation. 
The function h defined by (3.28) differs but little from its value slX p = q = 0, 
namely (3y, and (3.29) becomes 



(5.1) 



2/? 



[ dp f dq y^yWip, g)/4 

J _ 00 «.' — 00 



2/5V'(f Cr, 2/)). 



Here we have used expression (2.5) for the mean square value of /(.r, y) and 
have assumed yq^/h in (3.29) to be negligibly- small in the region where W{p, q) 
is different from zero. The average value of the reflection coefficient for hori- 
zontal polarization now becomes 

(5.2) 1 - 2fSV- (fix, y)). 

A similar treatment of (4.3) which involves the neglect of p" yb shows 



STEPHEN O. RICE (S367) 367 

that (5.2) also holds for the case of vertical polarization. It is interesting to 
note that (5.2) agrees with the first two terms in the expansion of a result ob- 
tained by W. S. Ament in which (3f is not required to be small, namely, that 
the roughness reduces the amplitude of the average reflected waves by the factor 
exp {— 2/3V(/^(x, y))]. As pointed out in the introduction, this agreement is 
all that the approximate nature of our results will allow. 

When W(p, q) differs from zero only in the region around p = 0, g = 0, 
equations (3.30) and (3.33) show that for horizontal polarization 

(I E, - 2i exp {-i^ax] sin ^7^ \') - 4/3V(.f(x, y)), 
(5.3) 

(I E. \') = Mflix, y)) 

where /^(x, y) — df(x, y)/dy; equations (4.4) show that for vertical polarization 

{\E^- E: r) = i&Vifix, y)) 

(5.4) (I E. r> = ^a\fl{x, y)) 

(I E. - e: P) = ^0VAn^, y))- .. 

Suppose now that W{ip, q) is sach as to make the terms yq^/b and p^/yh, 
which were neglected above, the dominant terms in the integrands of (3.29) 
and (4.3). The average distance between hills of the corresponding surface will 
now be small compared to a wavelength. The magnitudes of the average re- 
flection coefficients are then approximately 

1 - 2^ I dp I dq yq'W(p, q)/ib = 1 - ys, 
(5.5) 

1 - 2(3 [ dp \ dq p'Wip, q)/4:yh = 1 -- 

<•' — 00 l* — CO / 



for horizontal and vertical polarizations, respectively. Here Sp and Sh stand for 
small quantities, and 7 is the cosine of the angle between the 2;-axis and the 
reflected ray. The remarkable thing about the reflection coefficients (5.5) is 
that they depend on 7 in the same way as do the corresponding reflection co- 
efficients, computed from Fresnel's formulas, for a good, but not perfect, plane 
conductor. 

For vertical incidence 7 = 1, a = and the two expressions given by (5.5) 
reduce to essentially the same thing, the q^ in the first expression (where the 
incident E is parallel to the y-Sixis) goes over to the p' in the second expression 
(where the incident E is parallel to the x-axis) because of the difference in the 
assumed incident waves. 



368 (S368) ELECTROMAGNETIC WAVES 

6. Propagation along Surface 

As the condition of grazing incidence is approached y approaches zero, and 
expression (4.3) giving the average reflection coefficient for vertical polarization 
breaks down. In this case a modification of the method used to study reflection 
may be used to obtain a solution corresponding to a wave guided by the surface. 
A solution of this sort is to be expected since it has been known for some time 
that a corrugated or slotted surface will support a typical "surface wave" in 
which the field decreases exponentially with distance from the surface. 

To start with, we take the perfectly conducting surface to be 

(6.1) z = 2P cos sx = f 

w^hich show^s that / is now merely a function of x. Guided by the known prop- 
erties of surface waves, we assume that there exists a w^ave in which the electric 
intensity is predominantly in the z direction (approximately normal to the 
surface) and that there is also a small component of £" in the x direction (in the 
direction of propagation). We also tacitly assimie that the velocity of propaga- 
tion of the wave does not differ much from that of a wave traveling freelj' in 
the mediima above the surface; i.e., if the propagation of the principal part of 
the wave is described by exp {ict)t — ihsx} then hs approaches (3 = 2ir/X as the 
amplitude P of the corrugations approaches zero. 

When we attempt to express our assumptions as equations some experi- 
mentation suggests the forms 

E. = J2AMh+ m,z), E, = 

m 

(6.2) 

E^ = E(h, z) + T. CMh + 771, z) 

where the summations wdth respect to m extend over all integers from — oo to 
oo and Am , Cm are small quantities which approach zero with P. In order to 
fix the amplitudes of the various components Co is taken to be zero so that 
there is no term corresponding to E{h, z) in the summation part of E, . Here 

E{h -\- m, z) = exp { —i{h + m)sx — ih{h + ^njz] 

(6.3) ( W' - {h + mysT', ^' > {h + m)V 

h{h -{- m) = \ 

{-i[{h + m)V - fY'\ ^' < {h + m)V 

so that the components (6.2) satisfy the wave equation. Since the difference 
between (3 and hs is assumed to be small it follows that h(h) is to be regarded 
as small. 

Since we do not intend to carry our approximations beyond 0(0^ f) we 
may use the first of the boundary conditions (3.10) which, for our surface (6.1), 
becomes 



d 



STEPHEN O. RICE (S369) 369 

E, = N,E, = -f,E, = 2PssmsxE, 
(6.4) 

= Psi-ie'"'' + ie-''')E, , 

This relation must be satisfied at 2 = 2P cos sx = / to within an accuracy of 
O(P^). 

Upon substituting the assumed expressions (6.2) for E^ and E, in the 
boundary condition (6.4), using relations of the form 

EQi -{- m, f) = E{h + m, 0)[1 - ih{h + m)/ + • • •] 

EQi + m, 0)/ = PE{h + 7n - 1, 0) + P^(/i + m + 1, 0) 

in the same manner as in the reflection problem, and equating first order terms 
we see that 

A','' = Psi, A'll = -Psi 
(6.5) 

^1'^ =0 if m?^ lor -1. 

The divergence relation div E = gives 

(h + m)sAm + h(h + m)C^ = 0, m 5^ 
(6.6) 

hsAo + 6(/i) = 0, m = 0. 



Since Ao is zero, 6(/?,) is smaller than a first order term (it will be shown later 
to be OiP^)). From the first of equations (6.6) it follows that 



Ci'' = -(h + l)sAl''/h{h + 1), C'H = -{h - l)sA'l\/h{h - 1) 
(6.7) 

Cl'^ =0 if m ?^ 1, Oor -1. 

Equating the second order terms in (6.4), and using (6.5) and (6.7) gives 
A^'' = iP[b{h + 1)A['' - sCi'' + hih - 1)A'H + sC'll] 

= p2 J /?/ - /3' + hV hs' -h ^' - hV] 



b{h + 1) b{h - 1) J 

(6.8) A','' = iP[h{h H- 1)^{^^ + sCi''] 

A'll = iP[h{h - 1)A'1\ - sC'll] 

Al^^ = if m ?^ 0, 2 or -2. 
The expression for ^0^^ is of particular importance because when it is combined 



370 (S370) 



ELECTROMAGNETIC WAVES 



with the second of equations (6.6), which we write as Aq^^ = —hQi)/hs, we 
obtain an equation which may be solved for the propagation constant hs in the 
.T direction: 



(6.9) 



h(Ji) = P'hs' 



hi - 



' + K's' hs' + (3' 



h{h + 1) 



b{h - 1) 



tVl 



This expression shows that h(h) is O(P^) and therefore, when P is small in 
accordance with our assumptions, hs is nearly equal to (3. Replacing hs by /3 
in (6.9) gives 



(6.10) 



bih) = -iP'0'sKl + 2fi/sr" + (1 - 2/3/s)-"1 



which shows that if s > 2,5, hQi) is negative imaginary and E{h, z) decreases 
exponentially with increasing z. Thus in this case we have a true surface wave. 
When s is much greater than ^ so that the surface has many corrugations 
in one wavelength of the electromagnetic wave, we get from (6.10) 

5(/i) = -2^P'/3's 

/is = ^ + 2P'/3's' 

and the principal part of the field is the surface wave 

(6.11) E, = exp {-^^(1 + 2PVs')x - 2P'(3'sz] 

which travels a little more slowly than a free wave. 

The same type of analysis may be used to investigate the surface wave 
which is guided by the more general rough surface described in Section 2. We 
assume 



(6.12) 



E^ = ^ AmnE{m + h, n, z) 

mn 

Ey = Yl B^JE{m + h, 71, z) 



E. = E{h, 0, 2) + E C^Mm + h, n, z) 

mn 

where the summations extend over all integral values of m and n between plus 
and minus infinity. Coo = 0, and E{m + h, n, z) is defined by (3.2) and (3.3) 
with m replaced by m + h. The situation is somewhat similar to putting 7 = 0, 
o: = 1 in the vertical polarization case of reflection. The boundary' conditions 
are 

(6.13) E^ = -UE, , E, = -f,E, 

and these, together with the condition div E = 0: 



STEPHEN O. RICE (S371) 371 

a{m + h)Amn + anBmn + ^(^ + ^^, ri)C„in = 0, m, n p^ 
(6.14) 

a/iAoo + h{h, 0) = 0, m = n = 

lead to the expressions 

A^^l = iamP(m, n), B^^l = ianP{m, n) 
(6.15) 

C^mn = —ia\vfi + wk + n\P{m, n)/hQi -\- m, n) 

for the 0(/) terms in the coefficients. The 0(f) terms in A^n and B^^ are 

A^^: = E ^'[«(^ - ^)C'h^ + 6(/i + k, l)AiV]P{m - k,n - I) 

ki 

(6.16) 

B'll= E »■[«(» - OC-Il' + 6(/i + fc, l)BiV]P{m - k,n-l). ■ 

kl 

Since A^J^ is zero, from (6.15), AqV is given by the second of equations 
(6.14). Equating this to the value of A^o^ given by (6.16) leads to 

(6.17) h{h, 0) = X) ahk{^' " ah' - a%k) \ P{k, I) \yh{h + k,l). 

kl 

As the roughness decreases, b(h, 0) approaches zero and ha approaches (3 
and we have 

h{h, 0) - - E a'^'k' I P{k, I) \yb{k + ^/a, I). 
kl 

When this is averaged over the universe of rough surfaces mentioned in Section 2 
and when (2.3) is used we obtain, upon letting L approach infinity, 

(6.18) hih, 0) - - f dp f dq^yWip, q)/4h, 

J —CO t' — CO 

where 5i is the function of p and q obtained by setting a = 1 in expression (3.28) 
for h: 

( Ifi' - (p + 0r - qT' 

(6.19) fe, = < 

(-^•[(^) + /3)^ + 9^-/3T^ 

The principal part of the surface wave is 

Eg = exp {—iahx — ib(h, 0)z} 
which leads us to introduce B = ib(h, 0) so that 

B' = -13' + a-Ji' = {ah - (3)2^ 

ah ^ (3 + By2(3. 



372 (S372) ELECTROMAGNETIC WAVES 

We may therefore summarize our result by saying that the principal part of the 
surface wave corresponding to the general (sHghtly rough) surface of Section 2 is 



where 



E, = exp {-1,3(1 + 5V2^')x - Bz\ 



B = B, + iB, = -i3' I dp j dqp'W(j), g) '45i 



and the attenuation in the x direction is —BrBi 3 (nepers/meter). The defini- 
tion (6.19) of 61 shows that Bi is never positive. It also shows that if Wijp, q) 
is zero where hi is real,, namely inside the cncle of radius 3 centered at p = — ;3, 
q = 0, Bi is zero and there is no attenuation. This corresponds to the case 
where the hills of the surface are close together and is in agreement with the 
\'iew that the guiding action of the surface is due to rapidly undulating com- 
ponents of 2 = }{x, y) while the attenuation is due to the scattering produced 
by the more slowly vandng components. It should be remembered that (6.18) 
is only an approximate expre^ion for h{h, 0). It seems probable that more 
accurate expressions would show an attenuation even if W{jp, q) were zero in 
the circle mentioned above because this is no guarantee that A I,, and Bl^^ given 
by (6.16) will vanish for values of m and n which correspond to waves carrjing 
energ^' away from the surface. Thus it appears that even though the surfaces 
z = P cos sx and z = Q cos tx can cany^ surface waves without attentuation 
when -s > 23 and t > 23. the same is not true of the surface z = P cos sx + 
Q cos tx if, for example. .5 — t were almost equal to 3. The situation is somewhat 
similar to the one encoimtered ia the discussion of reflection from the surface 
(3.36). 

7. Reflection from Wavy Interface betiveen Tivo 
Media — Horizon tal Polarization 

Let the interface coincide approximately with the plane z = and let the 
propagation constants a and r of the upper (2 > 0) and lower media,, respectiveh*, 
be given by 



(7.1) a = ico(^£eo) ' = i3, r = a{er + g icoen) 



1/2 



Here we have assumed that both media have the same permeabHity jj. and that 
the ratio of their dielectric constants is e, . g is the conducti^-ity of the lower 
medium and cq the dielectric constant of the upper mediiun. The upper medium 
is non-conductiag. For free space n = 1.257 x 10~° henrj-^ meter and €0 = 
8.8.54 X 10"'' farad meter. 

If the interface coincided exactly ^^ith the plane z = the electric intensity 
for horizontal polarization would be 



(7.2) 



STEPHEN O. RICE (S373) 373 

Ey = E"^ = exp {-(jQfrcKexp \<jyz] + R exp {—ayz}), 2 > 

E, = E- = T exp { -aax + tt'^}, z < 
ra' = ao; = iav, 7' = (1 - a'^)^/^ 



i2 = - ^^ 



1 + ir i + ir 

<7'7 (77 

As before, a = sin and 7 = cos 6 where 6 is the angle between the 2;-axis and 
the reflected ray. 

When the equation of the separating surface is z = f{x, y) = / we assume 
the electric intensity to be 

!X) AmnE{m, n, z) for z > f 
X] GmnFim, n, z) for z < f 

(E^ + Z B^Mm, n, z) for z> J 
(7.3) Ey = < 

\E- + Z iy^n^(^, n, z) for 2; < / 

( E C'«.n£'(m, n, 2;) for z> f 

\ IZ IranFim, U, Z) lOV Z < } 

where E^ , E~ are given by (7.2) (with the dividing surface z — replaced by 

z = }{x, y)) and 

E{m, n, z) — exp { —ia{mx + ny) — ih{m, n)z} 

F(m, n, z) = exp { —ia{mx + ny) + ic{m, n)z} 
(7.4) 

i6(m, n) = 1<t' + a'(m' + n')f'' a = 27r/L 

^c(m, n) = [r' + a'(m' + n')]'''. 

Here 6(m, n) is the same as the b{m, n) defined by (3.3) and^is either positive 
real or negative imaginary. The same would be true of c{m, n) if the lower 
medium were non-conducting. 

At 2 = / we require the continuity of 

E. - NXN^E^ + N^E, + N^E:) 
(7.5) 

Ey - NyiN^E^ + NyE, + N^E,) 



{ 



374 (S374) ELECTROMAGNETIC WAVES 

and two other expressions obtained by substituting H (the magnetic intensity) 
for E. When we assume the components N^, , Ny of the normal to be small (so 
that N, ~ 1), and also assume E^ and E^ to be small, (7.5) becomes 

E, - N,N,E, - N^E, 
(7.6) 

. (1 - N';)E, - N,E, . 

The H conditions corresponding to (7.5) and the assimiptions that N^ , 
Ny , E^ , Ey and their derivatives must be small tell us that the two expressions 



(7.7) 



dy ^^ ^^^^ dz ^^' dx ^ ^^' dy 
dz dx ^ ' "= dz " dx ^ ' dy 



must be continuous at 2; = /. Here we have made use of the assumption that 
the two media have the same permeability, and have neglected 0{f) terms. 
The terms Nl dEy/dz and NyN^ dEy/dz may be omitted from (7.7) since the 
first of the two relations 

(77(1 - R) = Try' 
(7.8) 

l-\-R = T 

ensures the continuity (out to 0(f)) of the terms in question. In the same 
way, the second of relations (7.8) enables us to omit N'^NyEy and NlEy from (7.6). 
When the assumed expressions (7.3) for the electric intensity are set in the 
boundary conditions (7.6) and (7.7), as just amended, the teiTQs arising from 
E^ and E~ can be simplified by using (7.8). For example, in the second of 
equations (7.6) these terms are 

exp {— o-Q;x}(exp {0-7/1 + R exp {— 0-7/J — T exp {Ty\f]) 
(7.9) 

= exp {-aax}fU-{-Oif) 
where 

(7.10) U = Tier' - t')/2. 

After similar reductions are made in (7.7), the four relations arising from (7.6) 
and (7.7) may be written as 

Z {[Ar.n + f.C^.Mm, n, f) - [(?.„ + fJm.]F(m, n, /)} = 
exp {-aax}fU + S {[B^„ + fyC^„]E{m, n, f) 
- [//.„+ /,/.JF(m,n, /)1 = 
(7.11j -exp l-aax] U[2f + ry' f] + i E {[-anCr^n + h{m, n)B^^ 



STEPHEN O. RICE (S375) 375 

— f.amBmn + f,anA,nn]E{m, n, f) 

— [-anlrr^n " c{m, n)Hmn — J,amHmn + f.anGmn]F{m, n, f)] =0 
X) {[-H'in, n)A^n + amCrr,n - LamB^n + /,anA J^(m, n, f) 

— [c{m, n)Gmn + amlmn — fyamHmn + fyanGmn]F{m, n, f)] = 

where 0{f) terms are neglected. 

We now assume a is such that aa = iav where v is an integer. In order to 
separate the first and second order terms in (7.11) we write the various coeffi- 
cients SiS A^ll + AZI + • • • , and so on, and use the approximate expressions 

E{m, n, /) = [1 — ih{m, n)f]E{m, n, 0) 
(7.12) 

F{m, n, /) = [1 + ic{m, n)f]E{m, n, 0). 

By replacing/ exp { —iapx} by its Fourier series expansion (3.16) and pro- 
ceeding as in Section 3 we find that the first order terms in (7.11) lead to 



J± ran — ^ mn, ^ mn — J^ mn 

(7.13) id{m, n)B'l\ - ian{C'Ll - l'l\) = 2UP{m - v, n) 

-d(m, n)A'i: + amiC'll - I'll) = 
where 

(7.14) d(m, n) = h{m, n) + c{m, n). 

The equations arising from the second order terms in the first two of equa- 
tions (7.11) may be simplified with the help of equation (3.17), the relations 
(7.13) between the first order terms, and the expansion 

(7.15) exp {-iavx]f = 2 P(/c - v, l)P{m - k, n - l)E{m, n, 0) 

where the summation on the right extends over all integral values of m, n, k, I 
from — CO to + CO. In dealing with the last two equations of (7.11) we need 
the additional results 

c^(m, n) — h'^{m, n) = a^ — t' 
(7.16) 

5(m, n)C':^n + c(m, n)l'l\ = 

the first of which follows from the definitions of c(m, n) and 6(m, n) and the 
second from subtraction of the first order terms in the two div E = equations 



(7.17) 



amAr„n + CLuBmn + h{m, n)C„„, = 
amGmn + CLuHmn " c(m, n)I,nn = 0. 



376 (S376) ELECTROMAGNETIC WAVES 

The results of this simpUfication are given by the equations 



(2) 
mn 



r(2) 



h 



B mn — H nn = hi 



(7.18) 



aniCZ - IZ) - Km, n)BZ - c{m, n)Hli = h 

amiCZ - IZI) - h{m, n)A^^i - c{m, n)G'li = K 
where, taking the summations over A; and I, 

h, = iam 2 {C'^i - Iki)P{m - k, n - I) 

h, = J^ [UPik - v,l) + ianidV - li\')]P{m - k,n- T) 

h = iYl [Ury'P{k - ^ Z) H- (d' - T^)B[\'\P{m - k,n -iT) 

h, = i{a' - /) X AiVPim -k,n-T). 



(7.19) 



Equations (7.13), (7.17) and (7.18) may now be used to obtain expressions, 
valid as far as 0{f), for the coefficients. From (7.16) 



(7.20) 



(1) 

mn 



K^n, '^) ^(1) ^(1) _ T(i) _ d{m, n) ^(1) 



c{m, n) ""' *"" "" c(w, n) 

and these relations enable us to derive the expressions 

, (1) ^(1) i2Ua^mnP{m — v, n) 



f(i) 

mn 



d{m, n)Dr 



(1) (1) i2UP(m - V, n) faV "1 

d{m, n) [.Dmn J 



(7.21) 



e 



fin) _ 

^ mn -^ mn j-\ 



-»■ mn 



i2Uan c{m, n)P{m — v, n) 
d{m, n)Dmn 

i2 UanPjm — v, n) 



where 

(7.22) 



Dmn = (i{yYi + ^^) + &(w, n)c{m, n). 

Explicit expressions for the /i's are obtained when (7.21) is put in (7.19). 
The second order terms may be obtained from 

dDA'li = a'm'hh, + {D - a'm'Xch, - K) + a^mn{hh^ - ch, + h) 
(7.23) dDB'll = a'n'bh + {D - a'n')(ch - h) + a'mn{hh - ch + h) 
dDCZi = T^a{mhi + nhz) + ca{mh4^ + nh^) 



Y 

i\ 



STEPHEN O. RICE (S377) 377 

by dividing through by dD (where we have written d, h, c, and D for d{m, n), 
h(m, n), c{m, n), and D^„). These expressions are obtained from (7.18) and 
(7.17) (written out for the second order terms). 

The manner in which these expressions approach the earher expressions for 
the perfect conductor may be examined by letting the conductivity g approach 
infinity. From (7.1) we see that, since a = i(3, r behaves hke a large positive 
number multipUed by i^^^. From equations (7.2, 4, 10, 14, and 22) 

a' = aa/r, t' = 1 + 0(r"') 

T = ^ + 0(0, U = -ayr + 0(1) 

T 

(7.24) cim, n) = -ir + 0{r~'), d{m, n) = -^r + h(m, n) + 0{r~^) 
D^r. = -irhim, n) + a\m' + n') + Oir'') 

iU/d{m,n) = ay -{- 0(1). 

In the case of perfect conductivity studied in Sections 3 and 4, one source of 
annoyance was the appearance of h{m, n) as a factor in certain denominators. 
Here the corresponding term is — irhim, n) in Z)^„ . Since him, n) may become 
small, or even vanish, we have retained the a^im^ + n) term in D^« . 
When T becomes large equations (7.21) become 

A (1) _ 2(Tya^mnPim — v, n) „ 

a (m -\- n) ~ irhim, n) 



(7.25) 



B'J: = 2ayPim - v, n\ ,, , ^ ff . ., r - 1 

\_aim-\-7i) — irhim, n) J 



2ayPim — v, n) 



^(1) 2(jyanP im — p, n) 



5(m, n) + ia (m + ^ )/t' 

When him, n) is very small a^im^ + n) is nearly equal to — o-^ and we may re- 
place the denominator in C^ii by 

(7.26) him, n) + ia\m^ + n)/r = him, n) - icr'/r = him, n) + i^''/r 

which never vanishes since him, n) is either positive real or negative imaginary. 
Thus the difficulty encountered in Section 3 (and, presumably, also that in 
Section 4) may be overcome by taking the electrical properties of the reflecting 
surface into account. 

The average value of E„ in the upper medium, from which the average 



378 (S378) 



ELECTROMAGNETIC WAVES 



value of the reflection coefficient may be obtained turns out to be the average 
value of 



(7.27) 



E^ + BlVEiv, 0, z) = exp {-aax][ex^ Uyz} + exp {-ayz](fi + B'X')\ 
d(p, 0)BlV = c{v, 0)/i2 - h 



^^"^ ^ div, 0) ir\ d{k, I) ID,^ \ 



— try 



X ir'Wjr - a^, s) 



where we have used the relations 

ic{v, 0) = Ty' 

r = ak 



av = (3a = aa/i 
s = al. 



27rk/L, 

When we let L approach infinity, the double summation may be replaced by a 
double integration in the usual way and we get, after some reduction, 

i2ay(a' - /) r°° , r°° ^ W(r - (3a, s) 



(B^i^) 



(7.28) 



(ry' + ay) 



L '^ L 



[- 



try 



-)] 



' c + h \r + s" + 6c 
where c and h denote functions of r and s defined b}^ 

ic - (r^ + r + sy 

(7.29) 

^6 = {a' + r' + sy = a^' - r - sT\ 

As Q approaches infinity (7.28) should approach the value of its counterpart, 
given by the double integral in (3.26), which was obtained in Section 3 for re- 
flection from a perfectly conducting but slightlj^ rough surface. That this is 
the case may be verified with the help of expressions (7.24) which hold for large 
values of g. 

Bibliography 

1. Feinberg, E., On the propagation of radio waves along an imperfect surface, Journal of Ph\"sics 

(USSR), Volume 8, 1944, pp. 317-330; Volume 9, 1945, pp. 1-6; Volume 10, 1946, 
pp. 410-418. 

2. Blake, L. V., Reflection of radio waves from a rough sea, Proceedings of the Institute of Radio 

Engineers, Volume 38, 1950, pp. 301-304. 

3. Carlson, J. F., and Heins, A. E., The reflection of an electromagnetic plane icave by an infinite 

set of plates. Quarterly of Applied Mathematics, Volume 4, 1946, pp. 313-329; Vol- 
ume 5, 1947, pp. 82-88. 

4. Lord Rayleigh. Theory of Sound, Volume II, Macmillan, London, 1929, pp. 89-96. 

5. Goldstein, H., Thesis, Massachusetts Institute of Technology, 1943. 

6. Brillouin, L., Wave guides for slow waves, Journal of Applied Physics, \'olume 19, 1948, 

pp. 1023-1041. 

7. Twersky, V., On the non-specular reflection of plane waves of sound, Journal of the Acoustical 

Society of America, Volume 22, 1950, pp. 539-546. In this article, published after the 
present paper was written. Dr. Twersk^^ has given an account of some of his work. 



i 



I 






i 



The Theory of Scattering of Radio Waves in the 
Troposphere and Ionosphere 



By H. G. BOOKER 

Cornell University 



Abstract 



Some success has recently been achievqi by Booker and Gordon [1] in apply- 
ing a theory of atmospheric scattering to explain certain phenomena of tropo- 
spheric propagation, and the same theory, with appropriate modifications, has 
now been applied to the ionosphere. It throws light on the phenomena of 
scattering in the E region, and could no doubt be used in connection with auroral 
phenomena and with scattering in the F region at times of ionospheric storms. 
The theory confirms an earlier suggestion by Booker and Wells [2] that the cut-off 
frequency for E^ is controlled more by the size of the fine structure in the E region 
than it is by the maximum electron-density. The scale I of the fine structure 
(as used in the theory of turbulence) is about l/(47r) times the wave length at 
which the Eg echo disappears. The strength of this echo should, according to the 
theory, be independent of wave length down to about the wave length 47r I, and 
its electric field should then decrease proportional to the square of wave length. 
The attenuation of the transmitted wave due to scattering should likewise be 
independent of wave length down to about the wave length 47r I and should then 
be proportional to the square of the wave length. To fit the theory to observa- 
tions, statistical departures of the electron density from mean are required that 
vary from a few per cent up to some thirty per cent. The maximum usable 
frequency for communication to a distance by means of E3 is, fortuitously, very 
similar to what would be expected if the cut-off frequency of Es were interpreted 
in terms of maximum electron density instead of in terms of the scale of the fine 
structure. 

It is an important consequence of the theory that disappearance of the E^ 
echo as the wave length decreases through the value 47r I approximately is due to 
a change from roughly omnidirectional scattering to predominately forward 
scattering, and that forward scattering is practically independent of frequency. 
This means that there is as much forward scattering at wave lengths less than 
10 meters in the E region as there is backward scattering below the cut-off fre- 
quency of Eg . Scattering in the E region (and probably also the F region) should 
therefore make an important contribution to fading of radiation entering the 
earth's atmosphere from cosmic sources. This fading will be most pronounced 
on cosmic sources which subtend an angle of less than X/(27r I), which is the angle, 
measured from the direction of incidence, within which the forward scattering is 

S379 



S380 



ELECTROMAGNETIC WAVES 



mainly confined. For such a source the ratio of the fading range to the mean field 
should be proportional to the wave length, and to the square root of the secant of 
the zenith angle of the source. Observations of the cosmic source in Cygnus by 
Bolton and Stanley [3] and by Seeger seem to fit in with the idea that an important 
part of the variation of this source is due to 'ionospheric twinkling." 



Bibliography 

1. Booker, H. G., and Gordon, W. E., A theory of radio scattering in the troposphere, Proceedings 

of the Institute of Radio Engineers, Volume 38, 1950, p. 401. 

2. Booker, H. G., and Wells, H. W., Scattering of radio waves hy the F-region of the ionosphere, 

Terrestrial Magnetism and Atmospheric Electricity, Volume 43, 1938, p. 249. 

3. Bolton, J. G., and Stanley, G. J., Variable source of. radio frequency radiation in the constella- 

tion of Cygnus, Nature, Volume 161, 1948, p. 312. 






i 



Properties of Guided Waves on Inhomogeneous 
Cylindrical Structures 



By R. B. ADLER 

Massachusetts Institute of Technology- 



Abstract 



An analysis is given of some basic properties of exponential modes on passive 
cylindrical structures, in which e, ji and a vary over the cross section and the 
bounding surface is not completely opaque. Major, but not exclusive, considera- 
tion is directed to lossless structures. Each mode is generally a TE-TM mixture. 
Some of the conventional orthogonality conditions do not remain valid. Condi- 
tions are discussed under which the instantaneous-, vector-, or double-frequency 
power flows along the structure are additive among the modes. Stored and dissi- 
pated energies generally are not additive. It is shown that the propagation 
constant for modes on a lossless structure cannot be complex; when the lossless 
structure has no confining boundary (like a dielectric rod) , the modes cannot even 
possess a true cutoff. Consideration is given to the relation between the direction 
of real power flow and that of the phase and group velocities. The frequency 
dependence of the field distribution is also interpreted. 

Further information on this material can be found in Technical Report 
No. 102, May 27, 1949, of the Research Laboratory of Electronics, M.I.T. 



S381 



,J 



Evaluation of Integrals Associated with Wave 

Motion in Dispersive Media and the 

Formation of Transients 



By MANUEL CERRILLO 
Massachusetts Institute of Technology 

Abstract 

(a) One basic research project of the Research Laboratory of Electronics, 
M. I. T. is the investigation and development of methods of approximate in- 
tegration of a class of integrals of the type 

(1) /(r, , • • • , T.) = 2^. £ F(S) exp lWis,r,,---, r.)] <h, 

particularly the subclass associated with transient phenomena in electrical net- 
works and other linear systems. 

Several methods have been investigated and developed. They can be 
classified into three basic groups: Open methods, Cliff methods and Pocket 
methods. This classification is in accordance with the way in which the contour 
of integration, y, encircles the singularities or passes in their vicinity. 

Other subclassification has been made depending on the method in which 
we approximate the function F{s), or W(s, ti , • • • , r J or both. These sub- 
classifications are: stationary phase, saddle point, extended saddle point, plain 
and mixed cliff, plain pocket, essential and substitutional methods. They are 
listed in the order of increasing generality. The last method is by far the most 
complete theoretically. It has been found that the substitutional method 
possesses powerful potentialities because it is a sort of generalization of the 
preceding ones. 

Investigation of this last method began about a year ago in a rather limited 
extent. Even at that time extensive applications of the method were foreseen 
in connection with the synthesis problem, etc., and as a consequence considerable 
attention has been given to the method during the last few months. 

(6) The basic ideas contained in this powerful method are outlined here 
in a condensed form. 

We are all aware that the singularities of F{s) and W(s, ti , • • • , rj, their 
branch cuts, etc., have a canonical role in the genesis of the integral formation. 
With certain conditions on the contour 7, which type (1) may satisf}^, it happens 
that the complete integral contribution comes directly from the neighborhood 
of the singularities of the integrand and along the banks of the cuts, around 
which we deform the contour of integration. 

S383 



S384 ELECTROMAGNETIC WAVES 

The theoretical experience with these integrals indicates that the net 
quantitative effect of the singularities on the integral solution depends strongly 
on their nature and more strongly still upon their relative position in the s-plane. 
Because of the relationship of the parameters, ti , • • • , t„ and the functions in 
the integrand, it becomes clear that these parameters influence, in general, the 
position, or sometimes the nature, of the above-mentioned singularities. 

In our integrals, the parameters ti , • • • , r„ possess a variable nature (inde- 
pendent, of course, of s). When they change, the singularities of the integrand 
may move in the s-plane, changing their relative position, or sometimes their 
nature, and therefore changing their effect on the process of the integral forma- 
tion. 

We may naturally wonder if this motion of the singularities of the integrand 
as a function of ti , • • • , t„ can be exploited as a possible basic idea for a method 
of approximate integration. To illustrate this possibility we must describe the 
process in more precise terms. 

Suppose that we are interested in an approximate solution of an integral 
contained in class (1), which will be valid in a certain domain of variation of 
Ti , • • • , Tn , say Gv .^ When ri , • • • , t„ vary in G^ , the singularities of the 
integrand will, in general, move in the s-plane. They follow certain orbits with 
a definite law of motion, thus changing their relative position. It may happen 
that during relative motion a certain group of singularities have, or may attain, 
an almost dominant quantitative control on the building up of the integral 
solution when ri , • • • , r„ are in G, ; while other singularities have, or ma,Y 
attain, a small or secondary quantitative effect on the integral when ri , • • • , t„ 
are in Gy . Let us assume momentarily that this is the case. 

The following notation is now convenient. Let ri , • • • , r„ be in G, . 
'Trimary" singularities (in Gy) are those which have a strong quantitative in- 
fluence and ' 'secondary" singularities (in GJ are those which have a minor 
quantitative effect on the integral solution. 

Now, let us follow the motion in the s-plane of the singularities of the inte- 
grand, with particular attention given to the primary ones. Let us trace in 
the s-plane the orbits and find their laws of motion,^ in terms of ri , • • • , r„ , 
for each singularity. 

The segments of the orbits, corresponding to the displacements of the 
primary singularities for ti , • • • , r„ in Gy , will occupy one or several regions 
of the s-plane, Dy . The union of these s-plane regions will be denoted by Dy . 



^We can think of Gy as follows: the n-tuple of variable numbers (n , • • • , t„) defines an 
n-dimensional vector space, say Rn , over the field of definition of the numbers n , t2 , • • • , r„ . 
Hence Gy is a subdomain of 72„ . 

2The reader may have a clearer picture of this process if we particularize the general case j 

by considering that n , • • • , t„ are, for example, continuous functions of the real variable t \ 

and set i 

Tk = Tk{t); k = 1, • • ■ , n 
in the interval ta < t < h{Gy is ^6 — to). The above equations define a line immersed in i?„ . 



f 



ABSTRACTS S385 

This situation clearly establishes a certain correspondence (not necessarily one- 
to-one) between G^, and one, or several regions of the s-plane, Z), . 

The final wave shape produced by the contributions of the primary singu- 
larities strongly depends upon the combined effect of every primary singularity. 
We may therefore look at them as a group rather than as isolated singularities. 
This group consideration is important particularly because: (a) For a given set 
and disposition of primary singularities, the relative displacement of one of them 
may cause, and this is often the case, a considerable change of the wave shape 
of the integral solution, (b) In some cases we can replace the group by some 
other analytical entities^ of simpler structure whose effect with regard to the 
integral solution is equivalent or almost equivalent, and vice versa. These 
entities sometimes define points in the s-plane which also move as functions of 
Ti , • • • , r„ . 

We will agree that when at least one of the primary singularities of a given 
group drops to a secondary rank, or as soon as at least one new singularity rises 
to the primary rank, then the given group must be considered as a different one. 

(c) Past experience with the approximate integration of a large subclass of 
integrals of type (1) has revealed and confirmed: 

(a) The reality of the primary and secondary singularities associated with a 
a given domain G^ , of variation of ri , • • • , r„ . 

(b) The raising or lowering in rank of certain singularities in different domains, 
say G^ and Gy+n . 

(c) The decisive influence on the wave form of the primary singularities when 
they are considered as a group rather than in their individual roles. 

fd) A basic structural composition of the primary group. For example, the 
improper modification of the group elements, say, by cancellation of one or 
several singularities, alteration of orbits or laws of motion, may produce an in- 
correct integral approximation, or a very slowly convergent solution. 

{d) In the light of all these observations, the following two questions arise. 
Let F%{s) and W% (s, ti , • • • , rj be two functions which satisfy the requisites: 

(a) They approximate respectively F{s) and W{s, ri , • • • , r„), i.e. 

F%{s)^F{s) )s£i), 

W%{s, Ti , ' • ' , Tn) -^ W{s, Ti , • • • , r„)) TktG, ; k = 1, ••' ,n. 

(b) The functions F%{s) and W*{s, ri , • • • , r„) contain the primary singularities 
(or corresponding analytical entities) respectively of F(s) and W(s, ti , • • • , r„) 
for Tk t Gy , k = 1, • • ' , n; s z D„ . 



^Among those entities are saddle points, branch cuts, window functions, etc. Inversely, 
;h cuts can be replaced by certain sequences of alternated poles and zeros. 



branch cuts can 



S3S6 ELECTROMAGNETIC WAVES 

(c) Suppose that the integral defined by 

exists for t;, t G, , k = 1, • • • , n. 

Questio?2s: 

I. Are the conditions a to c sufficient to assure that /* == / for Tk t G, , 
k = 1, • • • . n (but not necessarily for r^. ^ G,) within a certain set of small 
tolerances? Supposing an affirmative answer for I: 

II. If now the orbits of the primarj^ singular points of F and W and, re- 
spectively, the orbits of the singular points of F* and W% become closer and 
closer and the respective laws of motion become more and more nearly equal 
for Tk ^ G^ , k = 1, • • ' , 71, then, can we say that the set of tolerances mentioned 
in I becomes smaller and smaller for r;; e G, but not necessarily' for tj, % G^ , 
k = 1. • • • . ?-/? 

If one considers as ''primary'" every singularity of the integrand, then a 
positive answer maj^ be given to the above questions and the solution is indeed 
valid for the complete domain of variation of ri , • • • , r„ . But if one considers 
only the primarj^ group, disregarding or modifj'ing the secondary ones, then, 
the answer maj^ be positive or negative and the problem is open to a further 
investigation. Direct mathematical proofs are very hard to conceive and 
construct. 

In view of some indirect consequences and results, which are derived from 
pre\4ous, knoTMi theories of approximate integration, we are fully aware, at 
least in many specific but illuminating cases, that we may affirm the possibility 
of a positive answer. The illuminating cases above suggest the existence of a 
positive answer for many other cases. 

The above sequence of ideas serves as a possible base for the development 
of a method of approximate integration. The above idea^, however, are not 
directly constructive since: (1) Xo means are pro^-ided to find or test the primary- 
singularities corresponding to a given G, . (2) Xo methods are given to construct 
the fimctions F% and IF* . (3) It is presupposed that the integral 



^ \ F%{s) exp {^y%{s, r, , ••• , rj)c/^ 



2m 

exists, but no means are provided to perform the corresponding integration. 

An intensive research has been conducted by the author, particularly during 
the last year, in order to supply the constructi^'e means which are necessary in 
apph"ing the above ideas. The new method of integration along these lines was 
called "substitutional." It is fairly well advanced, although it is still far from a 
final goal. To illustrate this we give below a few examples of the approximate 
integration of a subclass of integrals. The results are striking for their simplicity. 



ABSTRACTS S387 

Let 

fit) = 2id} ^^^^ ^^^ '^^ ~ "^^^^^ ^^' n = I, T, = t, 



where 



J "^ J ' 



Co being the abscissa of uniform convergence. 

Let us consider solutions for small values of t in the cases: 

Case I: 

n (« ~ S;fc) n m 

F(s) = C -^ ; 0(s) = 0; let p = m — n; go = Z) ^^ — 2 «/ 

n (^ - h) 

1 
Case II: 



V n (« - 1) n (s - s.) 

f(s) = C . ; ^ ; <^{s) s 0; 

vn(s-i) n(s-i.) 

^ 1 1 

let p = 7 - 5 H ; go = Z) ii« - Z) Ljn, ; 

jjfc algebraic multiplicity of the pole or zero. 

Case III: 

F{s) as in cases I and II; </>(s) 7^0 but satisfying condition (/)(s) — ^ Ms^ with N < 1, 

S-»oo 

Let 0(s) be expanded around s = <» as 



(s) =8l^a„ + ^ + ^+ •••). 

General Solution (three cases) : 

fit) - U-^{T) exp { -a.r} ^ A(,_,)(X); < r « 1 

where: 

U-i{t) = unit step function 

A(p_i) = lambda function of order (p — 1) 



T = t — ao ; X = 2\^r{qo + Oo) 



k 



S388 



ELECTROMAGNETIC WAVES 



Case I 







Case II; ai = 

Note: one single-term solution for this large family of integrals. 

(e) Basic ideas of the particular methods of integration, which are men- 
tioned in section (a), as well as illustrative examples, were given in the S\Tn- 
posium presentation of this paper. A complete and detailed discussion of the 
subject will appear in a Technical Report No. 55, M.I.T., Research Laboratory 
of Electronics, 



f 



Electromagnetic Research in the U. S. Air Force 

Research Program 



By N. C. GERSON 
Geophysical Research Directorate, Air Force Cambridge Research Laboratories 



In a few pages, it is not possible to describe in detail a full program of 
mathematical investigations in electromagnetics which the U. S. Air Force would 
like to see pursued. However, some particular fields of interest may be briefly 
and generally indicated. 

At the risk of repetition, I would like first to state the Air Force policy with 
respect to research. The Air Force, in fostering studies in mathematical physics, 
does so with the clear conviction that today's research becomes tomorrow's 
practice, and that any policy which confines itself solely to present problems and 
their immediate solution soon leads to scientific degeneration and stagnation. 
It is true that the Air Force desires some applications of its sponsored research. 
However, the obtainment of maximum results requires that this view be liberally 
interpreted in the sense of forging tools for mathematical physics which will be 
available not only now, but also, if required, ten, twenty, or fifty years hence. 
In applying this policy towards investigations in electromagnetics, the Air Force 
finds itself vitally concerned with long-term studies, such as extension of present 
theory and clarification of anomalies. 

During the war, the necessity for mathematical investigations in electro- 
magnetics was obvious. The results were phenomenal. As one example, we may 
mention the advances made by Schwinger's integral equation and variational 
techniques, and the impact made by these on contemporary research in pure and 
applied physics. The celebrated success of concentrated investigations in mathe- 
matical physics, obtained by such groups as Radiation Laboratory, Massachu- 
setts Institute of Technology, and the Institute for Mathematics and Mechanics 
of New York University, are well established. The present Air Force sponsorship 
of electromagnetic research at New York University represents not only a con- 
tinuation of the wartime initiated studies, but also a marked expansion in their 
scope and objectives. 

In jointly sponsoring this Conference, both the AF Cambridge Research 
Laboratories and the New York University had but one objective in mind: the 
dissemination of recent advances in electromagnetics among the participants and 
the infusion of new vigor into current research. Because of recent progress in 
some aspects of the science, it becomes opportune to review at this time the 
advances in electromagnetics, and to examine the hypotheses upon which they 
stand. Additional progress may still be made in utilizing unexploited techniques 
for applied problems. It is wise to uncover the stumbling blocks in electromag- 

S389 



S390 ELECTROMAGNETIC WAVES 

netic research and to search for means to bypass them. From an open discussion 
of the results and pitfalls, new thoughts arise, and the general stimulation breeds 
further progress in this field. 

There are innumerable problems in electromagnetics still awaiting solution. 
It is the hope of the sponsors that a partial listing will stimulate interest in and 
promote thinking about these problems. Indeed, it is very possible for the 
researcher on the scene to overlook the obvious and become entrapped in more 
complicated solutions. It is hoped that the various problems outlined below ^ill 
germinate and sprout in the minds of some of the readers. 

The subjects of interest to the Air Force are not all being investigated at the 
present time; even if they were, reproducibility of results is desirable. The 
various research topics fall into several distinct but interlacing categories and 
may be described as: (a) development of broad mathematical techniques and 
procedures; (b) study of theoretical problems having present application; and 
(c) research on the foundations and rigorous development of electromagnetic 
theory. 

The first classification covers a thorough investigation of new mathematical 
devices needed for the solution of the various integral equations of electromag- 
netics. In order to make significant progress in any phase of activit}^, proper 
equipment is necessary; in electromagnetics, equipment is S3mon3TQOus ^dth 
mathematical techniques. In this connection, a considerable amount of addi- 
tional effort is yet required on phase integral methods; these methods ma}^ then 
be employed to obtain approximate solutions of the differential equations not 
only of wave mechanics, but also of propagation and diffraction. The W.K.B. 
method and its possible applications to partial differential equations should be 
more completely investigated. The usual W.K.B. procedure neglects reflections; 
however, by modified methods, solutions, <}>{x), may be obtained in terms of an 
infinite series of the type 

(t){x) = (^0 + 01 + 02 + • • • 

The first term, 0o , represents the common W.K.B. approximation; the next 
term, 0i , represents the distributed reflections ignored by the first term; the 
third term, 02 , represents the reflections produced by the second term, etc. Since 
the occurrence of reflections gives rise to forbidden zones, the W.K.B. method 
does not provide the zone structure required for periodic potentials. By con- 
sidering the first order reflection term, 0i , in the above series, however, a simple 
description of band structure may be obtained. Other extensions are needed 
particularly with respect to low values of electromagnetic frequencies and com- 
plex dielectrics. 

The widely known basic method applied and adapted by Schwinger emplo^^s 
variational techniques for the solution of a variety of physical problems. With 
this method the function that makes a given ratio of integrals stationary is the 
same function as the solution of a certain integral equation. The standard pro- 
cedure, which has been employed to obtain solutions of the vector wave equations, 



N. C. GERSON S391 

lies in (a) formulating the problem as a differential equation with specific boun- 
dary conditions; and (b) transforming it through the use of Green's theorem into 
an integral equation. By utilizing Schwinger's variational theorem and attempt- 
ing suitable trial functions in the variational problem, a solution may be obtained, 
and, in some cases, an estimate of the error involved. 

Obtaining the estimated error is a great advantage of Schwinger's work 
when compared to other methods of geometrical optics or of successive approxi- 
mations. The method has been employed to determine the diffraction of a scalar 
plane wave by an aperture in an infinite plane screen, and to study proton- 
neutron scattering at low particle energies. Application of variational techniques 
has been made to quantum mechanics in an effort to determine accurate values 
of the asymptotic neutron densities under given conditions. It seems reasonable 
to expect that Schwinger's variational techniques may be used advantageously to 
solve more difficult aspects of specific vector wave problems by means of dyadic 
Green's functions. 

Another route for exploration Ues in determining those coordinate systems 
wherein the scalar and the vector wave equations are separable. This general 
problem has been extensively studied by the pure mathematicians, and some 
progress has been made on the special problem of separating variables of the one 
particle Schroedinger equation in three-space, and in the separation of variables 
for the two particle wave equation. The fruitful results already obtained in 
seeking special coordinate systems wherein the wave equation is separable 
encourage further study. 

Extension of the Weiner-Hopf technique, required for the exact solutions 
of some diffraction problems, should also be undertaken. With this method. 
Green's theorem or a modal analysis is employed to represent the solution by 
means of an integral equation, which is then solved by a Fourier transformation 
and the process of analytic continuation. 

Additional mathematical techniques which hold promise may also be men- 
tioned. Studies on the expansion of solutions of Maxwell's equations in orthogo- 
nal functions may have useful applications in theoretical quantum electrody- 
namics as well as in theories of microwaves. This technique frees theory from 
the requirements that the field be expanded in plane waves. Another method to 
be investigated is that of determining alternative representations of Green's 
functions which, in general, lead to a more direct and useful solution of the wave 
equation. It is highly desirable on some occasions to represent the solution of a 
differential equation in terms of different series, each having a different domain of 
rapid convergence. 

With respect to the second category of interest to the Air Force, i.e., the 
host of problems which have some practical applications, only a few can be men- 
tioned; these few are in the field of propagation and scattering of electromagnetic 
waves. A very fertile field for increased investigative activity is presented by 
theories of diffraction. Diffraction by almost any object, including circular discs 
still requires considerable study, particularly with respect to edge effects in the 



S392 ELECTROMAGNETIC WAVES 

immediate vicinity of the diffracting object. Rayleigh pointed out that certain 
singularities occur at the sharp edges of the diffracting screen, a principle later 
demonstrated in Sommerfeld's solution of diffraction by a semi-plane. Diffrac- 
tion of electromagnetic waves around a sphere, considered rigorously by Mie and 
extended by Wilson, has recently been reconsidered by Foch. For his work 
Foch assumed that the transition from the illuminated to the shadow region on 
the surface of a sphere occurred in a narrow strip along the boundary of the geo- 
metrical shadow. These investigations, however, are but a beginning, and more 
work should be attempted. One problem of great complexity is that of diffraction 
by a dielectric wedge. The general problem is as yet unsolved although various 
approximations have been proposed. 

A little progress has already been made in treating diffraction by a random 
screen, but the topic as a whole, including diffraction by a sphere whose surface is 
periodically perturbed, has scarcely been examined. These problems are generali- 
ties of specific topics which arise in connection with electromagnetic propagation 
around obstacles, with radio wave propagation through meteorological frontal 
systems, and with scattering of radio waves by ionospheric or tropospheric 
inhomogeneities. Simple aspects of some of these topics have already been 
partially treated. 

The scattering of electromagnetic waves by dielectric discontinuities requires 
further amplification, especially for such problems as scattering by ellipsoids. 
Investigations must also be undertaken on the scattering of waves by plane sur- 
faces (such as hexagonal plates) , considering variously polarized waves as well as 
specific orientations of the plane of the scatterers. The problem should not only 
be considered for plane surfaces of constant size, shape and dielectric constant, 
but also under conditions wherein each of these factors is randomly distributed 
about a mean value. Scattering by means of spherical and spheroidal particles 
whose dielectric constant is a function of both space and time also requires exami- 
nation. These and similar problems arise in microwave meteorology in connec- 
tion with the scattering of microwaves by snowflakes and ice particles, both of 
which may melt as they fall to the ground. 

Propagation problems may by no means be neglected. In this group are 
those involving wave propagation (at appropriate radio frequencies) through the 
ionosphere, the troposphere, and through very highly ionized media, as, for 
example, may be found in stellar atmospheres. Propagation over rough surfaces, 
such as irregular terrain, or over rough time-varying surfaces, as over swaying 
trees or a disturbed ocean should be given more attention. With respect to the 
ionosphere, better methods of determining true heights of the reflection layer 
are desired. Solutions of the wave equation near the ''reflection" surface for 
very long radio waves refracted from a medium (the ionosphere) where the 
refractive index is a rapidly varying function of the coordinates are still lacking. 
Additional thought is also necessary on non-linear propagation in the ionosphere, 
as well as on wave propagation through a highly absorbing volume (exemplified 
by the auroral zone). 



N. C. GERSON S393 

The third category for study may be termed special topics in electromagnetic 
theory. The use of asymptotic expansions described at this symposium in the 
paper, ''Asymptotic Expansion of Electromagnetic Fields," by Dr. M. Kline, 
warrants considerably more extension. Increased studies are needed to determine 
appropriate singularities in the electromagnetic field near sharp, well defined 
corners and edges. 

It is perhaps appropriate to mention that in addition to Maxwell's classic 
equations, Gibbs' fundamental work on dyadic analysis has led to an adequate 
account of wave reflection and absorption in both isotropic and anisotropic media, 
and to correct results for dispersion and absorption. Although Gibbs' work is in 
his usual comprehensive style, his advances in this field have received somewhat 
scant attention; theoretical work utilizing his theorems still merits serious study. 

The reader may perhaps wonder at the Air Force's interest in such a large 
variety of electromagnetic studies. The first group of investigations is devoted 
to the tooling stage, i.e., to the derivation of techniques and devices; the second 
category considers applications; and the third encompasses special problems. 
Our interest in all groups arises from the fundamental premise, proven by the 
historical development of the physical sciences, that the research of today is the 
practice of tomorrow, and that unless we devote an absolute minimum of activity 
to long-range fundamental studies, we have no investment in the technology of 
the future. 

Research in the United States today represents an unbalanced program. 
Great effort is devoted toward experimental investigations whereas we lag dan- 
gerously in theoretical studies. Must we wait for the pressure of an emergency 
to develop our theories and techniques, or is it possible during normal periods to 
theorize and progress? Why can we not balance our effort, not by dropping our 
experimentation, but by increasing our theoretical efforts? Certainly there is 
much more to research than instrumentation or the immediate solution of 
particular applied problems. 



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